First of all, be sure the meter is set to read on the impedance (ohms) scale. Also be sure you are using the correct probes (there is usually different jacks to plug the meter probes into based on what you are trying to measure. They will be marked).
The impedance / continuity setting works by putting a voltage across the probes. There is a battery inside the meter that powers this function. Check or replace the battery if in doubt.
Be sure you set the meter scale to it's lowest setting (for example, if it offers a choice of 0~10 ohms, 0~1000 ohms, and 0~10Kohms choose 0~10 ohms) if you want to check for continuity (testing a fuse element). If you want to measure the heating element, cycle through the settings until you get a value less than infinity, or until you are using the highest possible setting.
Now, since it beeps when you touch the probes together, everything should be as it should be ... I only stated the above in the interest of being comprehensive. What is the reading on the meter during this test? Whatever it is, that is your "zero" reading (eg it might be perhaps 0.05 or 0.20 ohms, consider whatever the reading is to be your zero value).
Closed (fuse OK) should be near or at your zero value.
Open (fuse fully or partially failed) should be some large impedance value (hundreds or thousands of ohms, etc).
It is possible there may be a resettable breaker instead of a go / no go "blowable" fuse. If that's the case, you may get an OK reading when in reality the breaker opens when powered up and then resets after the fault is removed, or goes through what is in essence a series of open / close cycles.
It is often important to measure while the Device Under Test (DUT) is out of the circuit. So, for example, to test a fuse, remove the fuse before testing.
All of the above is assuming you are using a Digital Multimeter (DMM) (which the unit you described is, so no problem).
But again, in the interest of comprehensiveness, the old style Volt-Ohm Meter (VOM) will damage most modern electronic devices. Only use such a meter on older gear, such as a Vacuum Tube powered radio.
A heating element is a direct short circuit, but one in which the resistance is high enough so it does not trip a breaker or fuse. So instead of blowing a breaker, the "wire" or heating element heats up instead of failing. So it may exhibit a large but not infinite and not zero resistance yet still be working as intended.