For those who submit obsolete GRE and IELTS scores (good scores), how likely is it for the admissions committee to reject/throw away the applications due to this? Will they instead review applications holistically?
GRE/IELTS scores are far from the most important part of your application. The faculty reviewing your application are likely to overlook this if the rest of your application is strong. However, the problem is that the applications may be filtered by computers or administrators before they reach the faculty (e.g., to remove incomplete applications) and they tend to be less forgiving.
A particular concern for you is that your scores will need to be verified at some point before you can register as a student. This requires you to get ETS to send the scores directly to the university. However, as far as I know, ETS will not send expired scores. Thus you will not be able to verify the expired scores and you will not be able register.
As annoying as these tests are, I suggest you re-take them to avoid the possibility of missing out on grad school for a silly reason like this. If you really cannot re-take the tests, you should try your luck, but I suggest that you make it clear on your application that the scores are expired. The worst thing would be for the university to think you have misled them.
This really depends on a million other things. If you're an outstanding applicant in every other way and this is the only issue then probably not. If it's a choice between you and candidate X who has everything in order, then yes - this would be an issue.
If you're applying in STEM (as opposed to, say, English literature), then this may be less of an issue: the school may make you take some internal English course to make up.
In any case, please do not lie in your application: there's no issue with submitting an old IELTS, but there is an issue with making it look like you just took it.