Quantum mechanics necessarily has infinities that have to be reconciled with. And some of the predictions there are some of the best tested and most accurate in physics. So to my understanding that seems like they are actual things that really exist.
Feynman diagrams are involved in a way of calculating the infinite possible paths that a particle can take to get from A to B. They are actually workable by humans because the more steps the less likely they are to happen, so if you calculate enough steps you get a very accurate answer. That seems to me like the infinity is involved in the actual underlying mechanics of how it works. And we just simplify it out when it becomes so small to not matter.
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