Do all those using New Keynesian models have to believe everything in those models? To answer this question, you have to know the history of macroeconomic thought. I think the answer is also relevant to another frequently asked question, which is what the difference is between a ‘New Keynesian’ and an ‘Old Keynesian’?
You cannot understand macro today without going back to the New Classical revolution of the 1970s/80s. I often say that the war between traditional macro (Keynesian or Monetarist) and New Classical macro was won and lost on the battlefield of rational expectations. This was not just because rational expectations was such an innovative and refreshing idea, but also because the main weapon in the traditionalists armoury was so vulnerable to it. Take Friedman’s version of the Phillips curve, and replace adaptive expectations by rational expectations, and the traditional mainstream Keynesian story just fell apart. It really was no contest. (See Roger Farmer here, for example.)
I believe that revolution, and the microfoundations programme that lay behind it, brought huge improvements to macro. But it also led to a near death experience for Keynesian economics. I think it is fairly clear that this was one of the objectives of the revolution, and the winners of wars get to rewrite the rules. So getting Keynesian ideas back into macro was a slow process of attrition. The New Classical view was not overthrown but amended. New Keynesian models were RBC models plus sticky prices (and occasionally sticky wages), where stickiness was now microfounded (sort of). Yet from the New Classical point of view, New Keynesian analysis was not a fundamental threat to the revolution. It built upon their analysis, and could be easily dismissed with an assertion about price flexibility. Specifically NK models retained the labour leisure choice, which was at the heart of RBC analysis. Monetary policymakers were doing the Keynesian thing anyway, so little was being conceded in terms of policy. 
So labour supply choice and labour market clearing became part of the core New Keynesian model. Is this because all those who use New Keynesian models believe it is a good approximation to what happens in business cycles? I doubt it very much. However for many purposes allowing perfect labour markets does not matter too much. Sticky prices give you a distortion that monetary policy can attempt to negate by stabilising the business cycle. The position you are trying to stabilise towards is the outcome of an RBC model (natural levels), but in many cases that involves the same sort of stabilisation that would be familiar to more traditional Keynesians.
This is not to suggest that New Keynesians are closet traditionalists. Speaking for myself, I am much happier using rational expectations than anything adaptive, and I find it very difficult to think about consumption decisions without starting with an intertemporally optimising consumer. I also think Old Keynesians could be very confused about the relationship between aggregate supply and demand, whereas I find the New Keynesian approach both coherent and intuitive. However, the idea that labour markets clear in a recession is another matter. It is so obviously wrong (again, see Roger Farmer). So why did New Keynesian analysis not quickly abandon the labour market clearing assumption?
Part of the answer is the standard one: it is a useful simplifying assumption which does not give us misleading answers for some questions. However the reason for my initial excursion into macro history is because I think there was, and still is, another answer. If you want to stay within the mainstream, the less you raise the hackles of those who won the great macro war, the more chance you have of getting your paper published.
There are of course a number of standard ways of complicating the labour market in the baseline New Keynesian model. We can make the labour market imperfectly competitive, which allows involuntary unemployment to exist. We can assume wages are sticky, of course. We can add matching. But I would argue that none of these on its own gets close to realistically modelling unemployment in business cycles. In a recession, I doubt very much if unemployment would disappear if the unemployed spent an infinite amount of time searching. (I have always seen programmes designed to give job search assistance to the unemployed as trying to reduce the scaring effects of long term unemployment, rather than as a way of reducing aggregate unemployment in a recession.) To capture unemployment in the business cycle, we need rationing, as Pascal Michaillat argues here (AER article here). This is not an alternative to these other imperfections: to ‘support’ rationing we need some real wage rigidity, and Michaillat’s model incorporates matching. 
I think a rationing model of this type is what many ‘Old Keynesians’ had in mind when thinking about unemployment during a business cycle. If this is true, then in this particular sense I am much more of an Old Keynesian than a New Keynesian. The interesting question then becomes when this matters. When does a rationed labour market make a significant difference? I have two suggestions, one tentative and one less so. I am sure there are others.
The tentative suggestion concerns asymmetries. In the baseline NK model, booms are just the opposite of downturns - there is no fundamental asymmetry. Yet traditional measurement of business cycles, with talk of ‘productive potential’ and ‘capacity’, are implicitly based on a rather different conception of the cycle. A recent paper (Vox) by Antonio Fatás and Ilian Mihov takes a similar approach. (See also Paul Krugman here.) Now there is in fact an asymmetry implicit in the NK model: although imperfect competition means that firms may find it profitable to raise production and keep prices unchanged following ‘small’ increases in demand, at some point additional production is likely to become unprofitable. There is no equivalent point with falling demand. However that potential asymmetry is normally ignored. I suspect that a model of unemployment based on rationing will produce asymmetries which cannot be ignored.
The other area where modelling unemployment matters concerns welfare. As I have noted before, Woodford type derivations of social welfare give a low weight to the output gap relative to inflation. This is because the costs of working a bit less than the efficient level are small: what we lose in output we almost gain back in additional leisure. If we have unemployment because of rationing, those costs will rise just because of convexity. 
However I think there is a more subtle reason why models that treat cyclical unemployment as rationing should be more prevalent. It will allow New Keynesian economists to say that this is what they would ideally model, even when for reasons of tractability they can get away with simpler models where the labour market clears. Once you recognise that periods of rationing in the labour market are fairly common because economic downturns are common, and that to be on the wrong end of that rationing is very costly, you can see more clearly why the labour contract between a worker and a firm itself involves important asymmetries - asymmetries that firms would be tempted to exploit during a recession.
Yet you have to ask, if I am right that this way of modelling unemployment is both more realistic and implicit in quite traditional ways of thinking, why is it so rare in the literature? Are we still in a situation where departures from the RBC paradigm have to be limited and non-threatening to the victors of the New Classical revolution?
 When, in a liquidity trap, macroeconomists started using these very same models to show that fiscal policy might be effective as a replacement for monetary policy, the response was very different. Countercyclical fiscal policy was something that New Classical economists had thought they had killed off for good.
 Some technical remarks.
(a) Indivisibility of labour, reflecting the observation (e.g. Shimer, 2010) that hours per worker are quite acyclical, has been used in RBC models: early examples include Hansen (1985) and Hansen and Wright (1992). Michaillat also assumes constant labour force participation, so the labour supply curve is vertical, and critically some real wage rigidity and diminishing returns.
(b) Consider a deterioration in technology. With flexible wages, we would get no rationing, because real wages would fall until all labour was employed. What if real wages were fixed? If we have constant returns to labour, then if anyone is employed, everyone would be employed, because hiring more workers is always profitable (mpl>w always). What Michaillat does is to allow diminishing returns (and a degree of wage flexibility): some workers will be employed, but after a point hiring becomes unprofitable, so rationing can occur.
(c) Michaillat adds matching frictions to the model, so as productivity improves, rationing unemployment declines but frictional unemployment increases (as matches become more difficult). Michaillat’s model is not New Keynesian, as there is no price rigidity, but there is no reason why price rigidity could not be added. Blanchard and Gali (2010) is a NK model with matching frictions, but constant returns rules out rationing.
 I do not think they will rise enough, because in the standard formulation the unemployed are still ‘enjoying’ their additional leisure. One day macroeconomists will feel able to note that in reality most view the cost of being unemployed as far greater than its pecuniary cost less any benefit they get from their additional leisure time. This may be a result of a rational anticipation of future personal costs (e.g. here or here), or a more ‘behavioural’ status issue, but the evidence that it is there is undeniable. And please do not tell me that microfounding this unhappiness is hard - why should macro be the only place where behavioural economics is not allowed to enter!? (There is a literature on using wellbeing data to make valuations.) Once we have got this bit of reality (back?) into macro, it should be much more difficult for policymakers to give up on the unemployed.
References (some with links in the main text)
Olivier Blanchard & Jordi Galí (2010), Labor Markets and Monetary Policy: A New Keynesian Model with Unemployment, American Economic Journal: Macroeconomics, vol. 2(2), pages 1-30
Hansen, Gary D (1985) “Indivisible Labour and the Business Cycle” Journal of Monetary Economics 16, 309-327
Hansen, Gary D and Wright, Randall (1992) “The Labour Market in Real Business Cycle Theory” Federal Reserve Bank of Minneapolis Quarterly Review 16, 2-12.
Pascal Michaillat (2012), Do Matching Frictions Explain Unemployment? Not in Bad Times, American Economic Review, vol. 102(4), pages 1721-50.
Shimer, R. ‘Labor Markets and Business Cycles’, Princeton, NJ: Princeton University Press, 2010.