Think about what bootstrapping does:
Bootstrapping estimates the properties of an estimator ... by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution function of the observed data. In the case where a set of observations can be assumed to be from an independent and identically distributed population, this can be implemented by constructing a number of resamples with replacement, of the observed data set (and of equal size to the observed data set).
Say that you are calculating a statistic from your data to estimate some parameter of the underlying population. Under the bootstrap principle, the distribution of the values of the statistic among your bootstrapped samples should be close to its sampling distribution in the population from which you drew your original sample. If that statistic has a normal distribution or a t-distribution in sampling from the original population, you might expect the distribution of the statistic among the bootstrapped samples to be close to normal or t.
If the distribution of that statistic is biased or skewed in sampling from the original population, however, it should be biased or skewed among your bootstrapped samples, too. For example, the plug-in estimate of the Shannon entropy from a finite sample from a population is inherently biased and skewed, leading to bias and skew in entropy estimates among bootstrapped samples.
That, however, is a major advantage of bootstrapping. You don't need to know the distribution of the statistic when sampling from the population: you let the bootstrapping itself give you an estimate of its distribution.