Absolutely do not leave the wires alone wrapped in electrical tape. Electrical tape does not have enough longevity to safely protect the exposed leads and can create a hazard in the future.
The blue wire exists to support the future addition of a light. The polite thing to do for your future self or future owner of your home would be to connect the blue and red wires. However, it is not necessary; if you do not want to, you can cap both leads with wire nuts.
Note that it is also legitimate to connect the black on the fan to the red from the ceiling, and the blue on the fan to the black from the ceiling. Assuming you have two switches, that will swap which switch controls the fan and which switch controls the light.
If the existing black-black connection doesn't work, you can try the black-red. If that does work, it would suggest that the switch for the fan has failed, although it could also indicate other wiring problems.
In a perfect world, with perfect electronics, there would be no need for a grounding conductor. It's there to deal with imperfections. The most likely case for a ground fault in this installation would be if the black conductor lost some insulation and shorted itself against the frame of the fan. In such a situation, the casing could become electrified, and a grounded person, touching the fan body could receive a shock.
If it were properly grounded, you'd get a short circuit to the ground wire, and the breaker would trip.
However, the fan is mounted to the ceiling, and the possibility of someone touching it is low. (Unlike, for example, the frame of a fridge or stove, where human contact is part of normal operation.)
So, Is it safe? Quite likely. For added safety, turn the wall switch off when changing bulbs. Is it to code? No. -- Unless, as others have suggested in comments, the mounting box is grounded on the other side.
You can however, locate the breaker, and replace it with a Ground Fault Interrupt breaker, which would trip should a ground fault occur. This isn't something you should do yourself unless you have experience working in the panel.
Another possibility, if your switch happens to be chained off of an outlet, which is quite possible, depending on the routing of the wiring on that circuit, is to replace the outlet with a GFCI outlet, and wire the switch to the load side of it.
Best Answer
I would exchange the fan. It sounds like it is either over heating and there is a safety feature that cuts it out or there is a problem with the motor winding and when it heats up it causes a bad coil wire or a bad connection to open.