Can a man on immunosuppressant cause their female partner to have urinary tract infections after sex?

Not really. An acute urinary tract infection can occur in females after sexual activity. It has little to do with the male partner taking immunosuppressive drugs.
Some drugs can... Be transferred through male semen. This includes immunosuppressive medications.
Yes. Vaginal intercourse is a risk factor for the development of utis in all women because of the close proximity between the urethral opening and the anus. Women can decrease this risk by practicing good hygeine and urinating as soon as possible after sex. Having an immunosuppressed partner, however, does not confer any additional risk.