This has been happening for about a week or so. I start the shower and the water is scalding - but within a minute or two it goes to lukewarm. I feel like if I were to stay long enough it would go to cold. A few times after the shower I tried the sink and it was hot and seemed to stay hot. This started the same time that we had a furnace problem, but the furnace people said that it was probably a coincidence. The shower faucets are the type where it is three dials, one for hot, one for cold and one in between that makes the water come from either the tub faucet or the shower head.

Furnace (hot air), or do you mean boiler (forced hot water or steam)? Hot water heater tank, or tankless on the boiler, or tankless separate?
When tankless on the boiler gets calcified deposits in it, slowly over the years, the first place you notice that less water can flow by the heating element fast enough is in the shower.

But you may also have crud in the hot water pipe to the shower from work done on the boiler. Take off the shower head and try a shower without it.

Hi,
Boiler with forced hot water. Tankless on the boiler. The water pressure is fine - does that make a difference? How can I clean out the pipes?