Is it safe to change the harddrive power feature so that it never turns off?

Yes, this is perfectly safe.

The advantage of powering mechanical drives down when not in use is simply to save electricity. Mechanical drives typically draw between 5 and 6 watts. You can look at your electric bill and see how much your power company is charging you per watt-hour and get an estimate on how much more it will cost you to leave it on. It isn't much.

Yes, it’s safe but it has its disadvantages.

1.- Shorter life span

Due to constant spinning mechanical parts will start to fail earlier. This is just how stuff works, things wear, tolerances broaden, things break. Most popular brands of hard drives will last a long time, if you take care of them. My thoughts on this, is have one small fast drive for the work and large drives for storage, which will be powered down most of the time. This way you use them efficiently.

2.- Damage to platters

If bumped when when mechanical parts are moving then you will risk irreparable, way too expensive, damage. Some laptops have an accelerometer that deactivates the HDDs, or move the heads away, when it's in free fall. I don’t know if this feature is available for desktops, or servers.

3.- Electric bill/ Laptop battery

It will draw more power, plain and simple. The power consumption will vary according to physical size, storage technology, etc; but in the end it is going to take its toll on your bill, or battery.

Do it if you need to, otherwise leave it be: it's just a couple of seconds.

Yes, this option is only designed to save electricity. In fact, spin-up/spin-down count is a much stronger indication of wear and tear than actual time spent spinning. Hard drives are designed to spin up and stay spinning; constantly spinning them up and down actually damages the drive more than keeping it spinning. See Is turning off hard disks harmful?