It might work as a antimicrobial (ie, applied externally) because it contains hydrogen peroxide but as to whether it can do anything about a cold (which is after all a virus) is another question.
The enzymes react with moisture in your skin to create hydrogen peroxide - read that somewhere. It also cuts off air flow to any bacteria. I think the sugar also acts against the critters.
As far as I know, honey has two material effects: it is tasty, and there are some studies suggesting it’s an effective cough suppressant. I’ve never heard of either one being temperature dependent.
I’ve heard people claim that local honey has anti-allergy properties. I would certainly believe that the active ingredients in pollen are destroyed at high temperatures. But AFAIK any purported benefits are entirely unsubstantiated, and the FDA notes that bee pollen can be actively dangerous to people who are allergic to the pollen.
Honey contains glucose oxidase from bee guts, an enzyme that produces hydrogen peroxide in the presence of glucose and oxygen[1]. The accumulation of small amounts of H2O2 is what makes honey shelf stable.[2]
It's probably also what causes honey to have antibacterial properties when you put it on a wound.
That's not to say that it would have any effect on bacteria in your body if you eat it.
The enzyme is deactivated or denatured somewhere in the neighborhood of 50 C. [3]
Honey itself has antibacterial properties but there's no evidence I can find to show those properties translating to your body after you eat it. It makes sense that honey is effective to help treat wounds because it's still honey, not mixed up in your stomach and beyond. I'd bet stomach acid would do a fine job denaturing that enzyme as well, heat be damned. So maybe don't boil your honey if you're going to put it on a scrape, but drink your tea as you like. I personally really dislike honey in tea.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3609166/