Pinched a wire, is this an electical danger?

There are two basic concerns

1. The electrical safety: As long as the wire is wrapped in electrical tape and not touching anything you should be fine. There are lots of wires in your house that have electrical tape on them. This is what electrical tape is used for.
2. The safety of the garage door opener: When something blew there may be damage that you are not accounting for. Test every safety feature of the garage door opener including but not limited to the safety eyes and auto reverse. Also, make sure the remotes and keypad/inside button still work. If you are satisfied with the operation of the undamaged systems then you can continue to use it.

If you damaged the wire then it is possible it is no longer capable of safely operating under the load that is called for. This is a potential fire hazard since it may result in the wire overheating.

You should replace the damaged part of the wire. Depending on circumstances of the wire (which you haven't fully detailed) you may be able to cut out the damaged section and splice the resulting free ends together (or splice a replacement section in to the gap). You should use proper splice hardware for the task, which could well be a simple wire nut (though this depends on the type and gauge of the wire as well as the service conditions - eg, wet or dry, exposed to oil, gas, etc).

If you are certain that the insulation and only the insulation was damaged (ie, the damage is no worse than would have been incurred had you been stripping the insulation from the wire with the appropriate tool), electrical tape may be a sufficient repair - again, considering service conditions.

It's hard to say whether the damage to the circuit board will result in dangerous conditions or not. Chances are, everything on the board is sufficiently low power that there is no danger - but with the information available here, we can't say for sure.

Any one that tells you that it is safe from this distance is a fool. Yes it may be safe, but you will need an expert to look at it.

After the repair, an electrical safety test should be done. This should be at minimum: checking the circuit diagram of the device, a visual inspection of the device, and a PAT test, or equivalent electrical safety test.

Call an electrician, they can do every thing that is needed to check that it is safe.