a few electric bikes have a throttle lever that you can use without pedaling.

Can it handle takeoffs without pedalling?

would the motor of most bikes handle this as well as hills that cannot be pedaled?.

The bike I'm thinking of is the e-joe Epik

The most common legal definition of e-bikes is that they are pedal-assist bikes, meaning you have to pedal for the motor to be allowed to power the bike.
That particular legal room also tend to restrict the power of the motor to 250 W. About 1/3 HP.
This means that if you only pedal enough to satisfy the conditions of the control system of the bike - or hack the controller - it can be kinda sluggish on climbs and takeoffs - depending on the weight of rider/bike and bike configuration.
If you use a pedal-assist e-bike an an e-scooter, it's probably entirely possible to overheat a motor. Whether the system is clever enough to recognize this and shut down before damage occurs or not I can't tell.
If you don't want to pedal, buy a scooter instead. It'll be built for that kind of use and likely to work for it.

If all you do is mash the throttle to max, and ride till the battery is drained, you may do some damage. Not only the motor, but also the controller. Both could over heat, but usually the controller has a safety shutdown to protect that component from excessive heat. I know of one Sondors that had a failure of the adhesive that holds the magnets in, from excessive heat. All you have to do is glue the magnets back in with a higher temp epoxy. My Sondors fat will peg at about 550watts with the throttle mashed. It has a rated 350 watt motor, but performs well at that 550 watts for a while. I have to use the throttle with pedal assist to basically get the bike away from a light with any speed at all. My weight, with a heavy single speed bike, requires it. I don't ride it like a scooter, I ride it like a bicycle with electric assist.

It's better for the motor if you assist it when "taking off" by pedaling, especially for smaller systems. The real damage to motors, batteries and controllers is when they're subjected to a high load and travel slowly (steep hill, high wind, large payload or whatever). If you don't want to pedal, get a bigger system.

+2 on 2olds reply.

High torque and low speeds will burn out a hub motor.

-SP

I built a mid drive e-bike on a dual purpose bike frame. I use the throttle a lot when riding in hilly country with no apparent damage, but I use it as an assist to pedaling. It makes getting started going up hill a lot easier and adds a little extra oomph to that very steep climb.