I see that an earlier message of mine is still waiting admin approval (probably because I quoted something from the QueueIt web site), so my comment to which you replied probably was too brief and vague. To put it briefly, RunDisney uses QueueIt to manage their queues and Haku Sports to process their registrations. Disney used QueueIt for the Guardians DVC preview last week, and in it was an explanation that anyone in the queue before opening would be sorted randomly upon opening. It's apparently their standard procedure, but I've never seen RunDisney acknowledge this. It makes so much sense of past registrations, though.
To answer your question, it does add a little bit of stress, but to QueueIt, and not to the registration process. I think it's best to think of QueueIt as a gatekeeper that's holding the hoards back. It's their responsibility to buy enough processing power to manage the queue expected, and they did a good job in this case. They seem to do the same for a lot of online queues, so I imagine they've carved out quite a niche for themselves, and really make pains not to screw things up. I would imagine it's not too data intensive, as what they seem to do is assign IDs to various browser configurations, hold them, then let them through the gates in a particular order. If a person were to have multiple queue IDs, there would be more browser instances, but as they're pretty much just counting and sorting, I don't think the strain is too much. In the case of those who are on the queue page before the doors are opened, they're randomized. Everybody who comes along afterwards are just sent to the back of the line. The registration servers would thus not be overwhelmed and crashed by thousands of people trying to get in all at once). It'll be set up to handle a particular number at a time, and will constantly be sending a message through to QueueIt to let through xxx number of more registrants.
I wondered if the delay this past Tuesday was due to server/processing power bought not by QueueIt to manage the line, but by Haku to process registrations. If either they or RunDisney didn't book this through AWS or another service provider, they wouldn't be able to process the thousands of registrations that they had per minute. It would also explain why they managed to fix it in a couple of hours - if this had been overlooked, it's not terribly complicated to buy some processing/server power, but it would take a little bit of time to configure everything correct. Total speculation on my part; it may have been that RunDisney limited capacity. We'll only know if and when they open registration and/or we see the numbers after the races are run next January.