Some opinionated points from my experience, doing mostly development and operations with only a bit of QA and support, for the past few decades. Make of them as you will.
I don't think it matters if bugs are picked up during formal QA, or by developers doing other work, or even customers or consultants. They should be treated, mostly, the same. My opinion is that they should all be recorded if they are serious enough that someone is motivated to point the bug out or report it.
Low priority bugs may later increase in priority (for business or technical reasons (e.g. other changes make the severity of the bug worse)).
Bugs can be incorrectly considered low priority and this can be noticed when e.g. support have a record that n people have called about a bug, or they have spent n hours due to the bug. If there isn't a ticket for support to find then they have to, in effect, create their own record to notice trends in calls and this work could have been saved. Support also don't always have the right perspective to accurately describe an issue in a way such that it will be found again when needed as they may only have a customers description - though that applies to everyone some of the time of course.
Working out what is wrong can take a lot of effort - more effort than a search for the symptoms (esp when you have a unique log message). Sometimes you observe something "odd" about a production system and you are unsure if its a symptom of something horrific. A quick search in the issue database to find that the problem is known and not serious can save a lot of time, especially at 4am after many stressful hours of recovering a system that encountered some mishap. I won't speak of the emotions of spending many hours of the night looking at an issue to find the next morning it was known but hadn't been logged and you could have had more sleep or spent time with your family.
It might not be obvious to everyone that an issue is low priority. People on-call may not be experts in every part of the system and may have just been pulled from their beds. They may well spend time diagnosing an issue and logging it only to have it closed. This is dispiriting for the reporter (speaking personally), especially in a product with a lot of issues. If there was already a record of the issue, while it may be disappointing that it wont be fixed, at least you haven't wasted time trying to write a good bug report and they might learn something (i.e. why it actually isn't a higher priority may give them insight into the system).
Sometimes developers don't have time/motivation for something big. Maybe they start holiday tomorrow and they just finished your planned work earlier than expected. They have a few hours now but don't want to start anything they can't finish or taxes their mind too much...they could just read the news till they think its safe to sneak out (so I've heard...) or maybe they could pick an easy but interesting bug and get some satisfaction of crossing something off. This requires an issue to be an easy fix as well as low priority of course.
A minor point but I have often used low priority bugs to train new developers on how the team works. There is no pressure on them to get the bug fixed by a particular time, or indeed ever, so they can concentrate on learning the SCM system, the build system, the ticketing system, whatever other archaic system inflicted on the dev team. If they manage to fix the issue it's a bonus.
Having lots of tickets shouldn't be a burden. If it is you need a better ticketing system or better method of recording of issues. Whether true or not, I have often had the impression that people who constantly worry about the bug numbers are never the people who actually fix the bugs, or the people arguing for more resources to fix the bugs, but are often the people who have to put the numbers in a power-point and pray no one asks them what they mean.
So my summary is record everything that has been found and described. It may help someone, someday, and the disk to store them is cheap. If you have people who are desperate for no open issues, suggest they fix a few bugs 
