Item 25 is not actually a thermostat, though it is listed as one. In reality it is a heater. It is used to add heat to the thermostat when using lower temperature settings. This has the effect of reducing the unit's heater running temperature.
Check it for continuity could be as high as 30,000 ohms. I do not have a wiring diagram so am unsure about the value many are around 7,000 ohms. It probably read open due to your meter scale being too low to read it.

Item 3 is the thermal cut-off (fuse)
You have to find out what caused it to go.
Note: that sometimes they do just blow on their own but changing it without checking other things is a gamble. I believe richappy (another regular poster on this forum) has done a study on thermostats and found a wide variation of actual trip point and what is specified.

Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run on high and the thermostats cannot regulate it.

If the above is OK then you will also have to replace the hi-limit as it should have regulated the temperature so the fuse did not blow.

You still have to find out why it blew.
Check that the belt is OK.
Check the seals (drum etc) in the unit. The air is pulled over the heating coils, through the drum and pushed out the exhaust. So any large seal leak will pull in room air and the cycling thermostat on the blower will run the unit hot.
Check that the lint filter is not coated with fabric softener residue which greatly reduces air flow.
Check/clean your vent system.
Check/clean the blower wheel.

If all OK you may want to replace the cycling thermostat as it's contacts may not be opening (welded shut).

[COLOR="Blue"]I've checked the heater (#25) again. It read about 3600 ohms. That's lower than what you suggested it might be - don't know if that is a problem.[/COLOR]
It sounds OK. It is just a heater so if it has continuity it is usually good. In any case it should not give you a no heat condition.

[COLOR="Blue"]I also checked the heating coil using your instructions. It's about 10 ohms across the terminal and was open from each terminal to the heater casing.[/COLOR]
Sounds good

I've checked the vent hose and it is clear. Stuck my hand into the fitting where it exits to the outside and that seems good too. And when the dryer is on I can feel a pretty good blast of air coming out of the vent to the outside.
Also sounds good.
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I haven't checked the belt, seals or blower wheel yet. I assume I could examine the belt and seals by taking off the top? Could you point me to hints on the best way to do that? I've seen conflicting instructions on how to get the top off. And is there an easy way to get to the blower? It seems to spin freely but perhaps there's a buildup of lint in it?[/COLOR]

Here is a good link with info on many Whirlpool models.
http://www.applianceaid.com/whirlpool-dryers.html
When up have it opened up you may as well check the rear rollers, front support rings, belt etc. basically an overhaul.
As to getting to the blower wheel I do not know of an easy way. The long filter chute has to be removed or undone and moved up to get at the wheel.

So far I would say you need a thermal cut-off and a hi-limit thermostat.

Although I removed all the stuff from the lint chute, I probably should have mentioned that it didn't seem to be obstructing the blower hole. However, I'm guessing that when the blower is on that some of it might have been blowing around and that might have caused a problem.

I've got one other question. I noticed that although the blower housing seal (#4) seems OK it is quite thin (probably between 1/16 and 1/8 inch). Is that the way it normally is?

Thanks for all the help. So far, I think I need a thermal cut-off (#3). I'll also replace the high-limit thermostat since it looks like they come as a pair. I'll also replace the lint screen housing seal since that is falling apart. I suppose I should get a new lint screen too since I assume that is at least part of the reason why there was so much stuff at the bottom of the lint chute. Would it be a good idea to replace anything else while I have the machine apart? I could replace the rollers. You mentioned that the cycling thermostat (#8) might be welded shut. I assume the only way to test that would be to heat it enough so it should open? I'm not sure of a reliable way to do that? If it is welded shut will the thermal cut-off go again? (Trying to decide if it makes sense to replace it with everything else.)

Out of curiosity, does it often happen that there is that much junk at the bottom of the lint chute? The machine is over 10 years old so I suppose that could be part of it. It looked like some of the debris was so small that the lint screen wouldn't catch it. However, even though the lint screen has a hole in it, it is very near the top so I would have guessed that most of the bigger stuff should have been caught by it.

check out my first post