We consider a general integrate-and-fire (IF) neuron driven by asymmetric dichotomous noise. In contrast to the Gaussian white noise usually used in the so-called diffusion approximation, this noise is colored, i.e., it exhibits temporal correlations. We give an analytical expression for the stationary voltage distribution of a neuron receiving such noise and derive recursive relations for the moments of the first passage time density, which allow us to calculate the firing rate and the coefficient of variation of interspike intervals. We study how correlations in the input affect the rate and regularity of firing under variation of the model's parameters for leaky and quadratic IF neurons. Further, we consider the limit of small correlation times and find lowest order corrections to the first passage time moments to be proportional to the square root of the correlation time. We show analytically that to this lowest order, correlations always lead to a decrease in firing rate for a leaky IF neuron. All theoretical expressions are compared to simulations of leaky and quadratic IF neurons.