Those ceramic "bullet" fuses are aluminum strips, and both corrode and fatigue crack. They often lose conductivity, and yes, loss of electrical power causes the heater valves to stay open all the time.
I'd get some good ceramic replacements (stay away from the plastic ones, they tend to "shrink" from heat) and replace all of them. A bit of dielectric grease won't hurt, either. Make sure the spring contact in the fusebox is clean for each fuse, too.
1972 220D ?? miles
1988 300E 200,012
1987 300D Turbo killed 9/25/07, 275,000 miles
1985 Volvo 740 GLE Turobodiesel 218,000
1972 280 SE 4.5 165, 000 - It runs!