In practice simulation is almost always done without full certainty about the distribution.
For example, one might consider a lognormal distribution for some random variable... but what if it was a little lighter tailed (gamma, say)? Or heavier tailed (inverse gamma, say)? Or much heavier tailed (inverse Gaussian, say)? Or log-t?
What if it was some mixture of lognormals? ... or perhaps some form of regime switching?
What if there was a slight dependence in the data?
One can consider a host of ways in which the model may be inadequate (preferably springing from an understanding of the process being dealt with), and see via simulation how that affects the conclusions.
Such things are commonplace. It's one reason why simulation is so useful
Think of the stock price return distribution-- it doesn't fit into a Gaussian Distribution, but what distribution we can use?
If you read around a little you can find a variety of reasonably good models; none will be perfect.
A relevant, but probably narrower question is, is there any research done along the line of sensitivity analysis? By "sensitivity analysis" I mean we assume known distributions for the variables, but we perturb them a little by slightly changing the distribution shape or slightly changing some of the variables, will the outcome of the simulation results remain roughly the same, or completely different?
Yes, in a variety of guises (and not just for simulation; it's a standard tool in robustness for example) -- but the perturbations needn't be small; you can investigate the effect of just about any form of variation from the base assumptions.
In robustness the effect of small perturbations in distribution is considered through tools like influence functions, empirical influence functions and sensitivity curves (and also via simulation)
The first several books at the above wikipedia page are a good starting place for references on robustness; I don't recall exactly what's in which book (I read Hampel et al in the mid 80's and Huber a year or two later, and some of the others since - it's been a while), but the Hampel et al book does discuss influence functions, so I'd start there I guess.
Edit: TooTone reminded me that I wanted to mention resampling methods, which can be thought of as a kind of simulation. For example, with simple bootstrapping, you use the ECDF (the cdf of the sample) rather than the assumed distribution to sample from. In more complex resampling schemes some function of the data (such as model residuals) may be resampled. So these approaches needn't rely on some parametric assumption about the distribution the data were drawn from.
(Then again, there's also the parametric bootstrap which might be as readily thought of as a particular simulation technique as bootstrapping.)
