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"I have a tear in my drywall against a door frame. "

Is it going t be covered by trim?

Tearing it off, priming (to help limit water from mud getting into the paper at the edges of the tear) and then using some mud is the usual repair.

You have to be careful that water dos not get into the torn edge and cause more blistering.

You can first cover the paqper with a product called GARDZ. It is intended for drywall tears but is a bit expensive for what you are doing. Another solution is to spray the ripped area with shellac. It will seal the area, then mud, prime and paint. Spray shellac is about $5

Just about any paint will wall paint put an adequate barrier up.

Gardz or oil prime first. Regular paint will bubble up more brown backing.