A PA34 PVT PLT ADMITS TO FAILING TO REMOVE A PITOT HEAD COVER DURING HIS PREFLT AT DAY; OH.

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I HAD FILED AN IFR FLT PLAN FROM DAY TO MIE; AS THERE WERE A FEW LOW SCATTERED CLOUDS IN THE AREA. AFTER RECEIVING CLRNC; I TAXIED TO THE RWY; PERFORMED MY RUN-UP CHK; AND CALLED THE TWR FOR CLRNC. THE TWR CLRED ME FOR TKOF; WHICH I ACKNOWLEDGED. THE TWR THEN TOLD ME THEY GOT A CALL FROM THE FBO; THAT MY PITOT COVER WAS STILL IN PLACE. I HAD NOT YET MOVED ONTO THE RWY; SO I ASKED TO TAXI BACK TO THE FBO. THE LINE MAN REMOVED IT FOR ME AND THE FLT THEN PROCEEDED NORMALLY. CHAIN OF EVENTS: MY NORMAL PREFLT WAS DISRUPTED BY MY BEING DISTR BY THE FUEL TRUCK WHICH WAS PARKED IN FRONT OF THE PLANE; AND MY CONVERSATION WITH THE LINE MAN. I HAD PREVIOUSLY ASKED FOR A QUART OF OIL TO BE PUT IN THE L ENG AND WAS CHKING WITH HIM THAT THIS WAS DONE. I ALSO WAS REPLACING DATA CARDS IN THE GPS RADIOS; WHICH I HAD JUST UPDATED. HAVING THE PLANE FUELED AND OILED; MY ATTN FOCUSED ELSEWHERE. HUMAN PERFORMANCE: I HAD WORKED THE NIGHT BEFORE; AS THE 'ON CALL' PHYSICIAN; THE JOB REQUIRING FILM READINGS AT HOME VIA TELERADIOLOGY. MY SLEEP WAS INTERRUPTED 3 TIMES; WHICH IS A LIGHT WORKLOAD FOR THAT JOB. I DID GET 3 HRS UNINTERRUPTED SLEEP; WAKING AT XA00 SPONTANEOUSLY. I FEEL THAT IN SPITE OF THE FACT THAT I FELT RESTED; MY PERFORMANCE WAS DEGRADED BY MENTAL FATIGUE; WHICH I WAS UNABLE TO ADEQUATELY ASSESS ON MY OWN. CORRECTIVE ACTIONS: 1) NO FLYING UNLESS I HAVE 8 HRS UNINTERRUPTED SLEEP. 2) ADHERE TO CHKLIST PROCS; ESPECIALLY IN NON ROUTINE OR DISTRACTING SITS.