To echo Aniko's comment: The primary assumption is the existence of truncation. This is not the same assumption as the two other possibilities that your post suggests to me: boundedness and sample selection.
If you have a fundamentally bounded dependent variable rather than a truncated one you might want to move to a generalized linear model framework with one of the (less often chosen) distributions for Y e.g. log-normal, gamma, exponential, etc. which respect that lower bound.
Alternatively you might then ask yourself whether you think that the process that generates the zero observations in your model is the same as the one that generates the strictly positive values - prices in your application, I think. If this is not the case then something from the class of sample selection models, (e.g. Heckman models) might be appropriate. In that case you'd be in the situation of specifying one model of being willing to pay any price at all, and another model of what price your subjects would pay if they wanted to pay something.
In short, you probably want to review the difference between assuming truncated, censored, bounded, and sample selected dependent variables. Which one you want will come from the details of your application. Once that first most important assumption is made you can more easily determine whether you like the specific assumptions of any model in your chosen class. Some of the sample selection models have assumptions that are rather difficult to check...