What fallacy is this? "Your argument is wrong/invalid because it's just an opinion."

I encounter this fallacy frequently in online discussions where an opponent completely disregards all of my premises and says my conclusion is invalid because it's an "opinion" and "not objective."

This is not merely questioning the bias of my sources: the opponent often makes no attempt to refute my premises. In extreme cases, they argue that only a 100% objective argument can be valid. I had this happen even in debates involving game design, visual design, or other disciplines/topics requiring subjective judgment.

I know from my critical thinking class that this is a fallacy because debates (by their nature) can be described as a clash of formulated opinions. But does this have a name?

One term for the fallacy you describe appears to be the subjectivist fallacy. From the Fallacies section of the Internet Encyclopedia of Philosophy:

The subjectivist fallacy occurs when it is mistakenly supposed that a good reason to reject a claim is that truth on the matter is relative to the person or group.

The proprietor of Logically Fallacious explains the subjectivist fallacy this way:

Claiming something is true for one person, but not for someone else when, in fact, it is true for everyone (objective) as demonstrated by empirical evidence.

The essence of the fallacy, it seems to me, is that it adopts as a working proposition the notion that if everything is relative—and therefore no absolute truth exists—it follows that no choice or opinion or course of action can be preferable to any other. Put another way, it rather mind-bogglingly argues that if we can see things only in relative terms, we have no grounds for making sound relative judgments.

It isn't a fallacy per se to dismiss an argument because it is a matter of opinion. It is, ironically, a matter of philosophical opinion whether opinions should ever be given any value at all. Positivism would say they should absolutely not:

Positivism is the philosophy of science that information derived from logical and mathematical treatments and reports of sensory experience is the exclusive source of all authoritative knowledge, and that there is valid knowledge (truth) only in this derived knowledge. Verified data received from the senses are known as empirical evidence. Positivism holds that society, like the physical world, operates according to general laws. Introspective and intuitive knowledge is rejected, as is metaphysics and theology.

(wikipedia)

Positivism has come into and out of favor over the years with very great minds both for it, against it and in between. If consistently applied, it is quite reasonable to dismiss an argument as just a matter of opinion if indeed it is and if all other arguments that are based on opinion are equally dismissed.

However, if the accuser is not applying the same rule to all other arguments (e.g. their own) then they are guilty of having double standards

a rule or principle which is unfairly applied in different ways to different people or groups.

(Google)

That could be called a fallacy if it was an honest mistake.