Depending on what your beliefs are, maybe nothing.. for now. However, if government or corporate interests decide that your beliefs are an issue for them at any point in the future, having them permanently stored could become an issue for you in the future.
But in doing so, one is talking about topics that one feels relatively safe about: either because one is expressing opinions that are in some ways culturally mandated, or which are so trivial one doesn't view them as controversial, or because one holds a social position whereby even though those views are contentious one feels they have enough social power to get away with expressing them.
But we don't have a perfect idea of what is deemed contentious in the future or what future contexts will be, so if you're not in a position of relative social power, and are assuming you're going to stay in that position for the plausible future, you're taking massive risks...or alternately your not really discussing very controversial opinions.
I'm lucky, in that I don't have anything that society currently views as a strong negative and I'm now in a relatively powerful position, but even I have to wear a "social mask" when I go out because my brain doesn't seem to work like most people's, and common topics of conversation dont really interest me.
But imagine others, or even me hypothetically investigating opinions I don't necessarily agree with. Indeed, back at university, for example, I attended various religious groups and meetings: I was interested from an anthropology perspective, but if someone had taken my name out and associated me with that group, that might kiss goodbye future employment.
And what about other groups and genuinely controversial opinions: gay men in religious communities, women in Saudi Arabia, minorities and ethnic groups, drug users, pirates, criminals, anarchy/communist/separatists/far right groups, fetishists and swingers, racists and religious fundamentalists (again, I did some work on the latter back at university). Even being interested in such topics in any serious fashion is enough to raise a bit of "social suspicion" from a social capital perspective...
Using aliases is rarely really anonymous, you can probably infer my identity pretty easily.
One can be above if one chooses, it's seems ultra-paranoid to want to discuss nothing unless you're anonymous - what do you talk about then in social situations IRL, do you wear a mask when you go out?