Nothing as dramatic as burning draft cards. Anyway, those people wouldn't have histories that survive. We do get a couple hints though.

I'm assuming you aren't referring to mutinies - these are often not objections to service in general, but rather a way of bullying the state into giving the soldiers a better deal.

The big one is entirely unattested. It's a bit of a hike to get there, so strap in.

The standard narrative of the 2nd century leading up to the crises of the Late Republic is usually framed around the Land Crisis and the Gracchan reforms. All the sources agree on the basics of the story: Ti. Gracchus got concerned with 1) the mass of urban poor, 2) the proliferation of slave labor inside Italy, 3) the lack of men that met the property qualification to serve in the army. Gracchus proposed to solve this by distributing publicly owned property, which Rome rented to wealthy men to farm (with slave labor) as a source of state revenue, to the urban poor, so that they would both reduce the need for slave labor on the big farms (latifundia) AND provide a new body of men of sufficient wealth to participate in the army. I'm going to skip over the the causes of this problem and success or failure of this project and it's long term effects for this question, but lots of us have written about it elsewhere.

That is the traditional narrative. All the sources agree. It seems to be entirely wrong.

Rosenstein recently wrote a book* Rome at War* (here's a review) that argues that there was no land crisis. There is no archaeological evidence of latifundia in Italy, or dissolution small farms, or proliferation of slaves. The sources are entirely wrong. So what did Gracchus think he was doing? He was responding to something, right? He wasn't manufacturing a crisis he died trying to fix out of whole cloth. Rosenstein suggests there WAS a dearth of soldiers, but not because of the property qualification. Rome's major engagement at this time was in Spain, which was a long, messy slog without much by way of loot. People just didn't want to go. They didn't show up when the call for soldiers went out, leaving the army understaffed.

I have a suspicion that this (whether right or wrong) is partly the product of a scholar who lived through the problems with the draft the US experienced in Viet Nam; if nothing else it might have inspired Rosenstein to look for another reason for the dearth of soldiers. We'll see in a generation or so how that works out.

There are a couple of other incidents.

After Punic 2, there was reluctance on the part of some centurions to serve in Macedonian 3. Livy gives the account at 42.32. Some people were being enlisted with a lesser rank than they held in P2, and the case was tried in the assembly. This isn't strictly speaking a refusal to serve, but rather service with conditions. Livy says the movement fell apart when they were shamed by another potential centurion.

The move towards paid professional armies might also be seen to follow this general line of protest. Scipio Aemilianus led an army largely of his clients to Spain.

There is somewhere a reference to the veterans of Scipio Africanus' African expedition being exempt from service in one of the wars following P2, but I can't find it at the moment. If I dig it up I'll add it.