Air Carrier First Officer reported being notified by a Flight Attendant of a 'burning odor' in the cabin during initial climb. Flight crew requested special handling to the departure airport. A safe landing was executed followed by the aircraft taxi to gate for an aircraft swap.

Narrative

I was assigned to the Relief Pilot position of Aircraft X; ZZZ-ZZZZ. I've known the other First Officer; since our days at Company; and had an excellent relationship with him. I had not flown with the Captain before but Name assured me he was a good guy; and I looked forward to the trip.Boarding was a mess because the cabin crew was shorthanded and there was a bit of conflict between them and the gate agents. After pushback we had a minor mechanical anomaly (Slats Primary Fail); and after discussing the issue with Maintenance Control and ZZZ Maintenance; we decided it was OK to proceed with the flight. During taxi out we received a runway change; then an amended clearance. At this point my 'Spidey sense' began to go off; noting that it was just like a sim session; with multiple distractions and problems. I mentally thought; OK; today is the day we get an engine fire on takeoff; so be ready.Captain Name1 made the takeoff; shortly after takeoff we received a Cabin Call annunciation. I answered the intercom; and FM Name2 informed me there was a problem with 4L; which I initially took to mean a passenger problem at seat 4L; but we quickly clarified that FA's (Flight Attendants) at both exits 4R and 4L were smelling a burning odor. I told FM Name2 I would call her back; and stand by. I always make a point of introducing myself to the cabin crew prior to boarding; and knew they were an experienced ZZZZ based crew that did not rattle easily or call false alarms; so I trusted Name2.I told Captain Name1 of the problem and suggested we return to ZZZ immediately. FO Name (Pilot Monitoring); suggested the same thing; and after a short discussion; Captain Name1 agreed; [advising] ZZZ ATC and requesting an immediate return. FO Name suggested he take over flying duties so Captain Name1 could run checklists and communicate with the ATC controllers and company; and control of the aircraft was transferred over to FO Name. I contacted FM Name2; and we briefed the return to ZZZ. I also advised the passengers that we had a minor system malfunction; and out of precaution we were making an immediate return to ZZZ; but there was nothing to worry about.I was receiving updated reports from FM Name2 during this time; first that the smell had dissipated; then that it was now being perceived at doors 3L/R. I communicated this to Captain Name1; who directed me to contact ZZZ Ops and advise them of our intention to return.By this time we were on a left base for Runway XX in visual conditions. Captain Name1 asked me to run the Overweight Landing Checklist. I initially could not find it in my iPad; realized I was searching in the FOM; not the AOM; shifted to the correct manual; and quickly located it.The chart directed me to locate and configure according to the Approach Climb Limit Chart. I had to quickly judge whether I should try to look up some numbers in a possibly detailed; unfamiliar chart and lose my place in the iPad; or monitor and backup the flying crew. The Overweight Landing Checklist directed that if 'Landing weight is greater than the landing climb limit weight' to use Flaps 20 and VREF 20 for landing. I knew that since Flaps 20 and VREF 20 was a good configuration for single engine approach; landings and go-arounds; it would probably be a good configuration/speed for our weight; so I told FO Name to use Flaps 20/VREF 20 for landing; and he reconfigured the aircraft; retracting the Flaps 30 he had already selected and reconfiguring the landing reference speeds for Flaps 20; VREF 20.At this point I realized how low we were. I knew that we were already going to a long runway and had no doubt we would be able to stop safely; so I elected to abandon the Overweight Landing Checklist; and backup the flying crew. We landed successfully; had the fire trucks inspect the gear; and taxied to the gate.During the debrief we were asked to prepare to move to another aircraft and redispatch the flight. I told Captain Name1 that I would not operate the flight; and needed time off to mentally recover from the emergency we had just been though. During a conference call with the Chief Pilot we all affirmed this fact; no pressure was put on us to continue; and the Chief Pilot agreed that we needed to rest and distress.Failings:-At no time do I recall; nor did I suggest; we run the Smoke; Fire or Fumes checklist. I will be kicking myself for this for a while; and can only ascribe it to the report from FM Name2 that the smell had dissipated; and I decided to prioritize the Overweight Landing Checklist. Nonetheless; I should have at least looked at the Smoke; Fire or Fumes checklist.- I had to quickly judge whether I should try to look up landing reference numbers in the Approach Climb Limit Chart; which I was unfamiliar with; or monitor and backup the flying crew. I elected to back up the crew and am also second-guessing myself about that decision.As Ross said in Fate is the Hunter; 'Anyone can do it when things are going right. In this business; we play for keeps.' I acknowledge that we did many things right; but we also could have done better on others. I look forward to feedback/training about how we (I) could have done better.