I don't think a hot shower has any substantial effect on body temperature, since this is carefully regulated by several mechanism to remain around 37$^\circ$C. When you expose the body to heat, you induce a response where the body increases heat loss and possibly reduces internal heat production in order to maintain the core temperature.
It is true that heating the body will increase blood circulation in the skin; this is a way for the body to carry heat from the core to the surface, where it can dissipate (radiate away). As for metabolism, I think the response to heat is less well studied than the response to cold, but presumably metabolic rate (specifically, mitochondrial respiration) will decrease in order to reduce internal heat generation, in order to maintain body temperature. Again, these are homeostatic responses that occur before there is any marked change in body temperature. Here is a review article on these mechanisms.
If you would expose the body to enough heat so that these normal responses cannot compensate, the body temperature will actually rise. Beyond 2-3 degrees (a normal fever), this is very dangerous because it will disturb a number of cellular processes. Cellular metabolism would certainly be affected, but mostly because some enzymes have narrow temperature ranges where they can function; I don't think the actual heat energy of metabolites is an important factor here, but I don't know of any study on this. A variety of other proteins and cellular structures will be similarly affected. This will eventually affect key functions in the nervous and circulatory systems, which is life threatening. The Wikipedia page on hyperthermia is a good starting point for more information on this.