Does the bitcoin URI message parameter have character limits? Is there a limit to how many characters are allowed in the message parameter of a BIP21 URI? I tried searching on the subject, but found nothing about it.
There isn't a limit described by the BIP (https://github.com/bitcoin/bips/blob/master/bip-0021.mediawiki). In general, however, URIs might be encoded into QR codes, so their length should be kept as small as possible. 

GET requests often have limits on the length of the encoded URLs, something like 2000 characters (https://stackoverflow.com/questions/266322/is-there-a-limit-to-the-length-of-a-get-request). This might be applicable depending on how you are using the URI.

As a guideline, however, I would say that messages in Bitcoin URIs should be fairly short. I'd probably try to stick to less than 200 characters.