I'm not talking about errors or omissions. Does a FP ever get rejected for example if there are too many flights scheduled into an airport or it conflicts with other flights or your FP as filed would take you through an active TFR area? How much scrutiny does a FP get when it's filed?
From this question I see that VFR plans get pretty much just filed away in case you need S&R. Does someone at ATC go through and verify that an IFR plan doesn't have conflicts?
Generally speaking, the only reasons for which a flight plan would be actually rejected would fall under what you referred to as "errors or omissions". But, the operational reasons you cite might lead to an amendment to your flight plan (which the pilot would most likely receive as an amended clearance).
IFR flight plans filed electronically are rejected by the flight plan processing computer if the filer files an impossible flight plan. I would presume that a flight plan filed verbally or in person through ATC or FSS would also be rejected for the same reasons, probably by the same computer, but with the rejection relayed to you through the third party. See these rejection examples that AOPA lays out in this article:
If you plan a "direct" flight, point-to-point, follow the rules and enter the correct equipment. For example, if "/G" (or "slash Golf," meaning that your airplane is equipped with an IFR-approved GPS and a Mode C transponder) or "/I" (you have loran or VOR/DME and a Mode C transponder) are not entered for these kinds of flights, the air traffic control computer will reject your flight plan because that route can't be flown with less capable equipment.
and,
File fixes that are actually on the airways that you file in your IFR flight plan. Pilot error causes flight plans to be rejected when the filed points are not on the routes that are filed.
Depending on how you file, other possible rejection reasons might include formatting errors, a non-existent clearance limit, or might depend on the specific filing means; I am certainly not familiar with all means of filing. I have filed in person with a sheet of paper when standing in the office at the bottom of a control tower, over the phone, and online, and over the aircraft radio; I've never had an IFR flight plan actually rejected.
Beyond such reasons, a flight plan will typically not be rejected as such, but may well be amended. The reasons you propose—traffic density, route issues, etc—are usually negotiable, or are circumvented by amendments to your clearance. Certainly, these issues may present challenges, but they can usually be worked around.
These flight plans do receive scrutiny; IFR clearances are all about traffic separation, and this is ATC's primary task. Understand that to file an IFR flight plan is, essentially, to apply for a specific routing for specific time frame; you may or may not be given your requested routing when you receive your actual clearance. Receiving a clearance different from the route filed, or receiving a clearance amendment while en-route are both common. Note that an amendment need not be ATC initiated: the pilot can request an amendment, either before or during a flight.
One final note on TFRs: you can sometimes (often?) fly through these, even under VFR; you just have to get approval. For example, I've flown through the Grand Forks TFR while flying VFR with ATC approval.
In busy airspaces like Europe and US east coast, checking for congestion is done. This does not mean directly comparing the specific routes and waypoints—conflicts on that scale are easily resolved by assigning different altitudes—but number of aircraft that are expected to fly through each sector, or land at each airport, at every hour. And only certain number is permitted so the controllers assigned to the sector can handle it.
But as far as I know the Network Manager Operations Centre (formerly Central Flow Management Unit; that does this in Europe) does not reject the flight plans, but it delays them. You file a flight plan and they'll give you expected release time. And if the destination or some sector along the route are congested, this might be several hours in the future. And then it's up to you whether you accept the wait or try to file a different plan that avoids the congested airspace. In airlines the dispatcher does this depending on how the extra fuel needed for longer route compares to the disruption caused by the delay.
