B737-300 HAD REPEATED CAPT'S AIRSPD-MACH INDICATOR FAILURES WITH OVERSPD WARNINGS. THE CREW RESET THE FAILURES FOLLOWING CHKLIST PROCS. A CONFLICT DEVELOPED ABOUT CHKLIST REQUIREMENTS AND MAINT PERSONNEL RECOMMENDATIONS. THE FLT CONTINUED TO THE INTENDED DEST AT A LOWER ALT AFTER CONCURRENCE THAT WX AND OTHER FACTORS WERE FAVORABLE.

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DURING CLB; THE CAPT'S ANALOG AIRSPD INDICATOR SHOWED AN OFF FLAG. AN ORANGE MACH FLAG AND THE OVERSPD AURAL CLACKER SOUNDED MOMENTARILY. THE FLT CLBED TO 13000 FT THEN 17000 FT DUE TO TFC AND THEN WAS CLRED TO FL280. WE ASKED ATC TO ALLOW US TO STAY AT 17000 FT. IN ORDER TO ASSESS THE PROB AND ADHERE TO THE EMER CHKLIST; SINCE THE FLAGS AND CLACKER WENT OFF AGAIN; THE FO FLEW WHILE I COMPLIED WITH THE CHKLIST AND RECYCLED THE CIRCUIT BREAKERS CALLED FOR IN THE CHKLIST. THE FLAGS RESET AND THE CLACKER STOPPED. AFTER BEING CLRED TO FL280 WE SENT AN ACARS MESSAGE TO DISPATCH ASKING THAT THEY CALL US. WE HAD DISPATCH AND MAINT ON THE LINE WHEN THE AIRSPD FAILED AGAIN AND THE CLACKER SOUNDED. I TOLD MAINT I WAS UNCOMFORTABLE WITH THE SIT SINCE WE HAD TO COMPLY WITH AN EMER CHKLIST THAT STATED AT THE END TO LAND AT NEAREST SUITABLE ARPT. WE LEVELED OFF AT FL240 WITH THE CLACKER SOUNDING IN ORDER TO STAY OUT OF THE HIGH ALT ARENA AND CONTINUED TOWARDS ZZZ1. WE RESET THE BREAKERS AGAIN AS PER MAINT'S REQUEST. THE AIRSPD INDICATOR FLAGS WENT AWAY AND THE CLACKER STOPPED. MAINT IMPLIED THEY WANTED ME TO KEEP THE BREAKER PULLED AND USE THE CAPT'S AIRSPD TAPE ON THE EADI. BUT I WAS UNCOMFORTABLE CONTINUING IN THIS WAY SINCE AGAIN WE HAD COMPLIED WITH AN EMER CHKLIST THAT GAVE GUIDANCE TO NOT RESET THE BREAKERS AND CONTINUE ON. I ASKED DISPATCH FOR A BURN TO ZZZ1 AT FL260 AND THEY WERE HELPFUL. THE ENRTE AND ARR WX WAS VFR AND; SINCE I HAD TAXIED FOR TKOF ON 1 ENG WITH A LONG TAXI DELAY; WE WERE AHEAD ON FUEL WHICH ALLOWED US TO CONTINUE AND LAND SAFELY IN ZZZ1 WITH APPROX 5000 LBS OF FUEL. I FELT THE COORD WAS GOOD WITH DISPATCH BUT FELT LESS THAN CONFIDENT WITH THE PROCS AND RECOMMENDATIONS FROM MAINT; ESPECIALLY WHEN I WAS TOLD ENRTE THAT THE ACFT COULD BE DISPATCHED WITH AN INOP AIRSPD INDICATOR IF WE USED THE TAPE. I ASKED FOR A PATCH TO THE DUTY FLT MGR. HE WAS VERY HELPFUL WHEN I STATED THAT; IF THE CLACKER WENT OFF AGAIN AND THE AIRSPD INDICATOR CONTINUED TO ACT ERRATICALLY; WE WOULD BE FORCED TO LAND AS PER THE CHKLIST. I ASKED FOR GUIDANCE OR INTERP. THE FLT MGR SUPPORTED MY POS AND STATED THAT WE SHOULD TAKE THE MOST CONSERVATIVE APCH. I WAS VERY COGNIZANT OF OUR PAX AND THE IMPLICATIONS OF LNDG SHORT BUT THERE WAS A SENSE OF PRESSURE COMING FROM MAINT. I REQUEST THE EMER CHKLIST BE REVIEWED ALONG WITH THE RELATIONSHIP BTWN AN INFLT PROB AND A DISPATCH DEFERRAL.