UPON TKOF; A B737-800 ACFT MASTER CAUTION LIGHT ILLUMINATED WITH A FWD CABIN DOOR LIGHT ON. TKOF CONTINUED; LIGHT EXTINGUISHED. DOOR HANDLE FOUND NOT FULLY CLOSED. ACFT DIVERTED ENROUTE.

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UPON TKOF OUT OF ZZZ AT 100 KTS THE MASTER CAUTION LIGHT ILLUMINATED WITH THE FORWARD CABIN DOOR LIGHT ILLUMINATION. WE CONTINUED THE TKOF AND THE LIGHT EXTINGUISHED. UPON LEVELOFF THE CAPT CHKED WITH THE #1 FLT ATTENDANT AND CHKED DOOR HANDLE DOWN. DOOR HANDLE COULD NOT BE SELECTED ALL THE WAY DOWN. WE COMMUNICATED WITH MAINT CTL AND THEY SENT A MESSAGE ADVISING TO NOT FLY UNDER THESE CONDITIONS BECAUSE THE DOOR COULD COME OPEN. WE DIVERTED TO ZZZ1 AND HAD MAINT LOOK AT THE SITUATION AND AFTER FURTHER REVIEW WITH MECHS AND MAINT CTL WE PROCEEDED TO DEST ZZZ2. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: REPORTER STATED THE FWD DOOR CLOSED AND LATCHED PROPERLY WHEN OPERATED AT THE DIVERTED AIRPORT. HIS COMPANY HAS EMPHASIZED A POLICY FOR THE PILOTS TO INFORM THE CABIN FLIGHT ATTENDANTS PRIOR TO TKOF TO RECHECK THE CABIN DOOR HANDLES FOR PROPER LATCHING. REPORTER ALSO STATED HIS CARRIER HAS HAD SEVERAL REJECTED TAKEOFFS) DUE TO CABIN DOOR WARNING LIGHTS COMING ON DURING TKOF; RESULTING IN BLOWN TIRES; OVERHEATED BRAKES AND ACFT DAMAGE. THIS IS A MAJOR SAFETY CONCERN AND FINANCIAL ISSUE FOR HIS CARRIER.