I frequently have to remind my members not to post their e-mails due to spam bots. One of them told me he doesn't care about that since his spam filter is good enough. If I allow him to do that, are there disadvantages to my website?
A small one.
Some spammers grab email address and domains to forge From: addresses. If this user's email is public, that might increase the odds that a spammer will forge with it, pretending to be sending email from your domain even though the spam is actually coming from someplace else. In which case, you might notice a slight uptick in warning notices to your maildaemon account, undeliverable email and the like. I presume you already receive a high volume of these warnings and are already filtering them appropriately.
Other than that, I can't imagine any possible downsides.
Further, the user is right, if you have a good spam filter and know enough not to fall for a phishing attack, the risk of publishing your email address are minimal.