My dryer won't shut off in any setting, and the heat does not come on. The control knob turns in all settings but fails to shut down once it reaches the off position. I have to open the door to turn it off. It is also not producing any heat in all settings. I have seen were people experienced one of these problems, but not both at the same time. I am going to start checking various parts, but any suggestions on what might be the problem would be appreciated. I don't want to start replacing parts if they are not causing the problem since some are expensive. This unit is about 10 years old so would I be better off just replacing it? Again any suggestions would be appreciated. Thanks.

The dryer is electric and I do have a meter. It may be a couple of days before I can take a look, but what should I check first? Thanks for the reply.

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Replacement parts for WHIRLPOOL LER4634JQ0 | AppliancePartsPros.com

See the attachment for the wiring diagram.

I am assuming that this unit used to work OK and it is not new to you.
If you did just get it, someone may have messed up the wiring.

Also I am assuming that the timer just keeps going past off in both timed dry and auto modes. The timer motor gets power through a different path depending on the mode.

Sounds like you have 2 problems.

No Stopping:
Unplug the unit and measure across the BK/BU set of timer contacts.
They should open when you are at off.
Be sure to disconnect the wires going to one side of the contacts, just to be sure you do not read an alternate/parallel circuit path.
Could be the contacts are welded shut.

No heat:
Measure across Timer Switch 2 contacts since you are already in the console.
They should be closed in all heating modes.
If OK then check the thermal cutoff / thermostats and then the heating coil.

If the thermal cutoff (Item 6 in Section 3) is blown, there are other things you should check before replacing it.

Thermal Cutoff blown.
Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 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 so the thermal cut-off blows.

The hi-limit should have regulated the temperature so the fuse did not blow, that is why there is a new one with the thermal cut-off..
Note: That unless there is another problem in the unit the hi-limit should never have to open. It is just a safety device with the fuse being a backup safety device.

Just in case it is not a grounded element.
With all the below the high limit will also have to be replaced.
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)