No. Doesn't matter whether handwritten or typed, so long as it's legible. Chances are, if it's an autograph request, the handler will just scan the letter, look for the item to be signed and the SASE.

I definitely don't mind my handwriting, but I find it difficult to write a handwritten LOR because the loops on my 'g's and 'p's run into the next line, which makes my letters difficult to read. Because of this, I normally send well thought out, but typed letters. Most of the time I don't have any issues when I do this (and have even received some letters back from stars complimenting my writing), but the few requests that I haven't gotten back have made me wonder if there are any celebrities who only respond to handwritten LORs. If you know of any, post them below so I and other autograph seekers know in advance! :)