Allowed or not, it's NOT safe. Quality tape is fine to cover bare wires, but it does NOT clamp the ends of twisted connections. Tape is only intended to cover bare wires, NOT to clamp the wires together. If any twist is not perfect, it will likely vibrate loose over time (or immediately), and start arcing. At least, this will cause light to flicker and electronics to fry. At worst, it could cause a fire. The wires have to be soldered, or clamped in another manner to maintain a tight connection. This electrician must be a real cheap son of a b****. Wirenuts cost 10 cents each (or less). How cheap can he be. I'd either insist that he use wirenuts, or find another electrician, and if you keep this one, the hourly wage spent nutting all the wires should be on him, not on you.
By the way, I often see electricians wirenutting wires without twisting them. This too is dangerous. Yes, I have done it for a temporary connection, but not permanent.