B737 Captain reported after takeoff the mode control panel would not accept any inputs. Upon reaching their destination they turned off all the automation and landed safely.

Narrative

Taking off from ZZZ; we were assigned runway heading on departure to 10;000 MSL. Prior to takeoff the Flight Directors were set to FO (First Officer) master; Autothrottle was armed; LNAV was disarmed and VNAV was armed. There were no warning flags on the PFD (Primary Flight Display) or ND (Navigation Display). After takeoff we raised the gear normally. At 400 AGL we anticipated the aircraft to transition to HDG SEL Mode. However it remained in TOGA. At 800 AGL we anticipated a thrust reduction and the assigned airspeed to jump to clean maneuvering speed. The assigned airspeed remained at V2+20. I tried to press the speed intervene button to manually set clean maneuvering speed but the MCP (Mode Control Panel) would not allow me to change the selected speed. I tried selecting Level Change; and the MCP refused to accept my input. I tried turning both flight directors off then back on; turning the FO side on first. Even with the Flight Directors off there was still a TOGA FMA (Flight Mode Annunciator) on the First Officer side but not on my side. The FO disconnected the autothrottles and we cleaned up the flaps on schedule. The Departure Controller assigned a vector direct to ZZZ VOR which was directly in front of us but after loading it into the FMC; we could not arm LNAV. Then once we reached 2500 AGL the TOGA FMA went away and we were able to engage the VNAV; LNAV and the Autothrottles normally again. There were no other anomalies throughout the flight until approximately 15;000 MSL on descent into ZZZ1.Descending on the ZZZZZ arrival with the Autopilot and Autothrottles engaged in ARM/LNAV/VNAV PTH; after passing 14-15;000 MSL; TOGA appeared on the FO PFD and the autothrottles added power and the aircraft began a climb. The FO turned off all automation and we reverted to using raw data. I informed the controller we had lost both flight directors and could not fly the RNAV RNP approach and requested vectors for a visual approach. I again tried cycling both Flight Directors OFF and then both back ON but the problem was not resolved. After reprogramming the FMC for a visual approach backed up by the ILS XXR; I noticed his assigned airspeed was different from mine . I tried a couple of other ideas to get our PFDs to agree by attempting to switch the CONTROL PANEL select switch to BOTH ON 1 then the FMC SOURCE select switch to BOTH ON 1. However TOGA indication remained on the FO side and we continued having assigned airspeed differences. I turned both flight directors off and left them off for the duration of the approach and landing however a flight director indication continued to appear on the FO side.We flew a visual approach to an uneventful landing on XXR in ZZZ and called Maintenance inbound to the gate in order to ensure we could give a full description to the techs before leaving. I called the Captain of the subsequent flight and briefed him and his FO on what we experienced. I told him that unless Maintenance was able to diagnose and replace the defective component(s); I would refuse the airplane.