I want to make a DIY antenna for my Kenwood TH-F6A. Is there any danger of damaging the radio (i.e. by trying an antenna that is unsuitable), or is it safe to experiment?
If you're making a simple, tried and true design like a dipole, quarter-wave vertical, or ground plane and you measure and build it reasonably carefully from properly calculated plans, there's absolutely nothing to worry about at the power levels an HT outputs. But that may not be what you mean by experimenting, really, so I assume you want to do more than that.
There is some danger of damaging any radio with an unsuitable antenna. "Unsuitable" can mean lots of things, but usually the damaging thing is that an impedance mismatch between the antenna and feedline or an antenna which is hopelessly far from resonant on the desired frequency causes too much of the transmitted signal to be reflected back into the radio. Again, a low power HT won't be putting out enough power to cause damage in most cases, and modern radios have some built-in protection against such damage, but to be safe with a truly experimental antenna you probably want to invest in an SWR meter, which gives an indication of the reflected power.
The main risk is damaging the radio with a high SWR. You can buy a meter that will measure the SWR, or your radio may have such a meter built-in. Or you can take precautions like reducing the transmit power.
That said, you should be relatively safe with an HT. The transmit powers involved are usually low enough that it's economical simply size the transistors in the power amplifier so they will not fail at any SWR. Moreover HTs are expected to operate with their antennas in proximity to other things (like the operator) which can cause large variations in impedance and thus SWR. As such HTs should be designed to be relatively robust against such faults.