By itself, no. But why was the tire replaced? If it wore out before the others, that indicates a problem most likely alignment and that could be the source of the burning rubber smell. If it was for something like a nail in an unrepairable area (sidewall, edge of tread), then that should not be the issue.
Burning oil does not smell like burning rubber. I don’t think the dealer did you any favors here by replacing the valve cover gasket. Sounds like he is firing up the parts cannon.
Look on the front of the engine and you will see a rubber belt that snakes around a bunch of pulleys. There is a pulley on the front of each accessory like the AC compressor, the alternator, power steering pump (if Honda is still using hydraulic power steering), water pump and the crankshaft pulley. There is also a tensioner pulley that keeps the belt tight and often an idler pulley that doesn’t do anything but guide the belt to the next pulley.
The accessory belt, AKA serpentine belt is the usual source of a burning rubber smell and it is most often due to a pulley that is not turning freely. Considering the age and number of miles on your car, there shouldn’t be any pulleys dragging. But sometimes a bit of road debris can get in behind a pulley and cause it to drag or it could get jammed in somewhere where it is touching the belt and making it hot.
The mechanic should have checked all this first and if needed, take the belt off the tensioner and spin all the pulleys except the crankshaft pulley and see if they turn freely and don’t wobble.
Edit: More about road debris. You could have picked up a piece of rubber or some plastics that is caught on a hot part of the engine, exhaust pipe, cat converter etc or caught in one of your axles and rubbing against the axle as it turns. They wouldn’t know that unless they put it up on the lift.