They are no longer on this shelf, but here is a closer look at the types I have. Perhaps they are too closely stuffed together?

Further detail.

Further detail.

Your main problem would be heat. Try & keep it in the low 60s during summer at the most. Probably do best in the 50s year round. Do not need any soil at all with these but using a freshwater sand washed in distilled water before use is probably better. Humidity at least at 50% though 60% to 70% may be better, depends on time of year. More humid in winter, less humid in summer, though it depends at what elevation you collected the mosses.
I would tend to put the tank with the mosses outside in full shade and spray with distilled water and take off the top so that wind/air would flow into it evaporating some of the water. Recover at night, leave outside if it would be safe from any animals that might get into it, including cats. Else bring it inside. Keep inside during your winters & bring in for a day or so at other times to inspect and look through the mosses to see how they are doing.
They should do OK if kept cool enough.

Thanks for your tips. Tap water not good, then? I will obtain a thermometer for heat and humidity and try to find a spot that works best. Outdoors is probably not gonna work, as it'll get brutally hot here in the summer. Would it be better to split them up? Any idea what kinds I'm dealing with? Sphagnum, I would guess, for most? I have no idea. I don't want to kill them.

It is hard to tell what they are as to ID but I cannot spot any Sphagnum. Really need close ups to attempt a sort out.
Most look like an open forest floor types of mosses. Some can handle heat some cannot. Depends on what kind of mitigating care is given.