The ball is dead. A-1 intentionally pushes B-1. Is this a technical foul or personal foul?

Dead ball = technical.

Dan is correct. If the ball is dead, then it must be a technical foul. The other question which the official needs to answer in this situation is whether or not to make this a flagrant technicalfoul which would lead to immediate disqualification. This is, of course, a judgement call by the official.

Dan,
Is the backside of a false foul a technical? ;)
mick

Mick,
It depends. The second foul in a false double or a false multiple foul could occur during a live ball, but before the clock has started following the first foul. This would then not be a technical foul.
Example: B1 fouls A1. Team A awarded a throw-in. Ball placed at disposal of thrower. B3 fouls A3 before the throw-in ends. False multiple foul during a live ball.
OR
B2 fouls A2. Team A awarded free throws. During free throw A4 fouls B4. False double foul, again the second is a live ball foul.

Or the second foul could occur during a dead ball and then it would be a T.
Example
A1 fouls B1 knocking him to the ground. B2 runs in and shoves A1. This is a false double foul. The second foul occurred during a dead ball so it is a technical foul.