When trying to fill my tub, the hot water only stays hot long enough to fill 1/4 of the tub. The rest of the water is lukewarm. At first I thought it was the water heater, so I filled the washer with all hot water and not once did the water become lukewarm. I also repeated this process with the bathroom and kitchen sinks. Maybe it is the water heater or maybe it is not. Does anyone have any idea of what may be causing this issue besides a faulty water heater?

I want to be sure I understand you correctly. Hot water elsewhere is not limited, just in the tub. The temp of the water to the tub drops after a minute or so? My first thought is your mixing valve but may need more info. Do you know how to shut off the main and remove the valve? Give that a try and inspect, clean or replace the valve if it is plugged. While the valve is removed cover the opening with a towel and have a friend turn the main on/off quickly to flush the line. Let me know if it helped.