B737-300 ACFT TEMPORARILY HAD SPD TAPE AROUND RADOME WHICH CAME OFF IN CLB AND ATTACHED TO THE CAPT'S PITOT TUBES. HIS AIRSPD INDICATOR DECREASED; BUT IN ACTUALITY THE AIRSPD WAS INCREASING. RPTR FO CALLED THE SPD DEV TO THE CAPT'S ATTN AND HE HAD FO TAKE OVER THE FLYING.

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WE WERE FERRYING AN AIRPLANE RATHER THAN OUR SCHEDULED PAX FLT DUE TO INBOUND WRITE-UP OF LIGHT AIRFRAME VIBRATION. A LCL MECH; AFTER INSPECTING THE AIRPLANE AND WITH DIRECTION FROM OUR COMPANY MAINT; PUT 'SPD TAPE' (DUCT TAPE) OVER ACCESS PANELS AND AROUND THE RADOME ATTACHMENT CIRCUMFERENCE. CLBING THROUGH APPROX 8000 FT TO OUR CLRED ALT OF 10000 FT WE HEARD A 'POP' SOUND. ALMOST SIMULTANEOUSLY (WE GOT A VNAV DISCONNECT AND THROTTLES WENT TO FULL THRUST) I NOTICED THE AIRSPD INCREASING QUICKLY TO 280 KTS. I QUESTIONED THE CAPT ABOUT HIS INCREASING AIRSPD WHILE STILL BELOW 10000 FT IN REGARDS TO THE 250 KT SPD RESTR. WHILE DOING SO HIS SPD WAS NOW APCHING PLACARDS. AT THIS POINT THE CAPT REALIZED HIS AIRSPD INDICATOR WAS NOT FUNCTIONING CORRECTLY (HIS AIRSPD INDICATOR WAS READING 160 KTS). AT THIS TIME HE PULLED THE THROTTLES TO IDLE AND XFERRED THE ACFT TO ME. I QUICKLY SLOWED THE ACFT DOWN TO 250 KTS. AS AN ASIDE TO THIS EVENT; LATER DURING A RECOVERING ATC CAUSED HIGH APCH TO LNDG; WITH THE CAPT'S AIRSPD INDICATING VERY SLOW; THE CAPT'S STICK SHAKER ACTIVATED THROUGH APCH TO LNDG. LATER MAINT INVESTIGATION DISCOVERED THAT THE RADOME TAPE HAD SEPARATED AND AFFIXED ITSELF TO THE CAPT'S PITOT TUBES (2). BE CAREFUL WHEN YOU HAVE TAPE FORWARD OF PITOT TUBES. SUPPLEMENTAL INFO FROM ACN 397943: WE HAD DECIDED TO FERRY THE AIRPLANE RATHER THAN OUR SCHEDULED PAX FLT DUE TO INBOUND WRITE-UP OF LIGHT AIRFRAME VIBRATION. A LOCAL MECH; AFTER INSPECTING THE AIRPLANE; AND WITH DIRECTION FROM OUR COMPANY MAINT HEADQUARTERS; PUT 'SPEED TAPE' (DUCT TAPE) OVER ACCESS PANELS AND AROUND THE RADOME ATTACHMENT CIRCUMFERENCE. CLBING THROUGH APPROX 8000 FT TO OUR CLRED ALT OF 10000 FT WE HEARD A 'POP' SOUND. ALMOST SIMULTANEOUSLY; MY AIRSPEED DISSIPATED FROM 250 KIAS TO AROUND 170 KIAS; THE VNAV DISCONNECTED AND THE AUTOTHROTTLES WENT TO LIMIT THRUST. I REDUCED PITCH TO REGAIN SPEED; COGNIZANT OF THE APCHING ALT CLRNC AND TRYING TO ASSESS WHAT HAD HAPPENED. MY FO QUESTIONED ME ABOUT EXCEEDING THE 250 KT RESTR BELOW 10000 FT AND I REALIZED MY AIRSPEED WAS UNRELIABLE. AIRSPEED WAS RAPIDLY APCHING PLACARD. I BROUGHT THE THROTTLES TO IDLE AND TRANSFERRED ACFT CTL TO THE FO POINTING OUT THE APCHING ALT LIMIT. AS AN ASIDE TO THIS EVENT; LATER DURING A RECOVERING ATC-CAUSED HIGH APCH TO LNDG; WITH THE CAPT'S AIRSPEED INDICATING VERY SLOW; THE CAPT'S STICK SHAKER ACTIVATED THROUGH APCH TO LNDG. LATER; MAINT INVESTIGATION DISCOVERED THAT THE RADOME TAPE HAD SEPARATED AND AFFIXED ITSELF TO MY PITOT TUBE! I WILL NEVER AGAIN ACCEPT AN ACFT WITH TAPE ON THE RADOME OR FWD OF THE PITOT TUBES.