The size of the sample has no effect on the probability of a type I error (rejecting a true null hypothesis). The probability of a type I error is dependent only on the significance level. Indeed, the probability of a type I error is equal to the significance level.
A Type I error is when the null hypothesis is rejected when it is in fact true. Increasing the sample size is a way to reduce errors, so I think you would be more likely to make a Type 1 error with a small sample.