B737 MAX 8 Captain reported the external communications door was not latched properly by ground crew resulting in a fairly loud; steady and annoying warning tone that could not be turned off. The aircraft flew normally with the door unlatched but the tone was exhausting for the flight crew.

Narrative

Aircraft preparation; checks; pushback; engine starts; taxi-out and takeoff were all normal. Aircraft was late pushing back (not sure of the reason). We took off on Runway XXR. We were assigned a left turn out of traffic to a heading of 280. Airborne and with a positive rate of climb; we retracted the landing gear normally. As we continued our climbing left turn to the assigned 280 heading and began retracting the flaps; on schedule; above 1;000 feet AGL. Once the flaps were completely retracted; a fairly loud; steady and annoying warning tone began to sound. We double-checked everything in the flight deck and could not find any additional warning or caution light. The aircraft was flying and climbing normally and all aircraft systems seemed to be normal; with the exception of the fairly loud; steady and annoying warning tone. We tried the landing gear warning and the altitude warning cutout switches; neither stopped the tone from sounding. The aircraft was pressurizing normally and re-confirmed during the 10;000 feet MSL Climb Checklist. I (Captain/pilot flying) put the aircraft on Autopilot and contacted the Cabin Crew (Flight Attendants) to see if they could hear the warning tone. They replied everything in the cabin was normal and they could not hear the warning tone. Via ACARS; I initiated voice contact with our Dispatcher. I explained our situation over the radio to our Dispatcher. He; in turn; conferred with Maintenance Control and we were told the most probable source of the warning tone was the external communications door was not latched properly by ZZZ Ground Crew following our pushback. We were even told ZZZ Operations reviewed our pushback and noted the Ground Crew did not properly close the external COMM door. We discussed a number of items; we knew the aircraft could be flown with external COMM door not secured. We asked how Dispatch wanted us to handle it; if we could no longer bear hearing the warning tone sounding (Flight to ZZZ1 planned at 2+47). We were told we could coordinate to return to ZZZ or land anytime along our route when the tone became exhaustive to us. First Officer and I discussed and we were comfortable continuing on toward ZZZ1 and monitoring each other. There was no applicable QRH Checklist to apply. We flew all the way to ZZZ1. As we were configuring to land on ZZZ1 Runway XYL; we had our flaps at 10 and assigned speed of 170 knots. When the gear was lowered and showed three green down/locked; the warning tone stopped sounding. We also received a number of ACARS alert tones (dings) which must have been inhibited while the steady warning tone was sounding. The remainder of configuring and landing were normal.We received a Master Caution and an overhead MAINT warning light as we were taxiing to our gate in ZZZ1. We were met by a ZZZ1 Maintenance Technician; who confirmed our external COMM door was closed/secured properly and who starting bringing up details of the MAINT overhead light and our warning tone situation on the First Officer's Navigation Display screen. The aircraft was labeled as an ETOPS aircraft; but I am unsure if it was ETOPS authorized. I closed the loop with our Dispatcher and Chief Pilot after arriving in ZZZ1.