B737-800 First Officer reported after takeoff and during descent both PFD's went blank and came back on when VNAV was engaged. During descent there was also a burning rubber smell in the flight deck.

Narrative

VNAV was an issue from the beginning of the flight. Upon rotation off of runway XX at ZZZ the captain stated that the vertical guidance in his HUD as well as his PFD blanked. The FO side blanked as well but both sides came back when VNAV was re-engaged. This occurred several more times in the climb; however VNAV stayed engaged when in level flight. The Captain and I discussed our ability to climb to and remain in RVSM airspace and the possibility of returning to the field. We both agreed that since it seemed to be a climb issue that we would be able to continue to ZZZ. Cruise portion on the flight was uneventful. As we began the descent on the arrival into ZZZ1 the issue again arose ; this time with much greater frequency. Each time the VNAV disengaged the PF re-engaged it. I thought about suggesting another mode of descent such as level change or VS and setting the altitude constraint in the window. However; I did not. As we were crossing energy I started to smell an acrid burning rubber smell. I asked the captain if he smelled it as well and he said he did. I opened the vents to see if it was coming from the packs because we couldn't identify where it was coming from. At that point I looked over and the captain already had the mask in his hand preparing to dawn it; so I followed suit. I began to run the QRC as the captain continued to fly and [requested assistance]. We were around 4 thousand feet at this point. The captain continued to fly and run radios; as I ran the QRH procedure. As I was running the QRH I heard the autopilot and auto throttles disconnect. I stopped running the QRH because it became more imperative to make sure the captain had the aircraft fully configured to land. Upon landing ARF followed us to the gate; we shutdown and took off the mask to find the smell had mostly dissipated. After speaking with maintenance we found the left side AOA vane had 57 faults through the duration of the flight. I believe the one glaring issue in this whole event was the repeated engagement of a faulty system the aircraft was continuously trying to tell us was broken. So the possible smoke and fumes was more than likely crew induced. I believe this to be caused by an over reliance on the VNAV system. There are multiple modes in which the aircraft will climb and descend in and it was certainly my failing for not suggesting using an alternate mode to the PF. A recommendation may be to incorporate a RNAV STAR with the VNAV inop during a recurrent event.