Can anything mimic prostatitis, and cause pain or burning in the scrotum and the penile shaft?

"Burning" pain usually means the skin is involved. Prostatitis does not cause such pain -- the symptoms are entirely different. Penile or scrotal skin pain might be a superficial infection or other dermatitis.

These are not the usual symptoms of prostatitis. You need to see a Urologist for a good exam and a Urinalysis. Prostatitis is acute and chronic. Acute is uncommon and makes you ill with fever etc. chronic can have many symptoms like pain in near the rectum, lower abdomen frequency, burning during urination etc.