Can it be normal to have back pain 4 weeks after spinal tap?

Possibly. It depends. If the "spinal tap" was done to get spinal fluid to test it and the "tap" was traumatic or difficult with the needle hitting bone you may have pain for a number of weeks. The outer layer of bone is called periosteum and has many nerve endings and can take some time to heal. However, the pain should not be severe. If continues get a work-up.
Spinal tap. No it is unusual, usually in causes pain at the place of niddle insertion which should not last for more than couple of days. In some cases some leak of csfoccur at the place of the tap and causing post spinal tap headache which can last from few days up to few weeks.