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As stated in the title (an IEEE conference). Does this create a conflict of interest antithetical to the reviewing process? Is it possible that accepting or refusing to review could jeopardise the chances of one's own paper being accepted?

No, there is no conflict. Your advice will be backed up or not by other reviewers. Likely the conference committee already knows that you are also a submitter.

But if you want double assurance of this, just send a note to the program chair that you have also submitted. If they see any issue, they will deal with it.

Just give an honest review as you would in any case. And trust that your paper will be accepted or not on its merits. Of course, there is quite a lot of competition, but that is always true.

I see no reason why it would be a conflict of interest. An analogous question would be if there is a conflict of interest in you reviewing papers in a journal you might publish in. Given that other conference attendees are in the same/similar field as you, and are interested in a good conference, those attendees would seem to be the ideal group to pull reviewers from.

In conferences where I've seen the sausage being made, accepting to review gains no advantage. Refusing any and all review requests will ultimately be viewed as rude, but will not disadvantage you in having your paper accepted. (Invitations for invited talks and membership on the conference committee may well be hindered, however.)

I've reviewed for many CS conferences and never seen one that has a conflict of interest policy saying that people who've submitted a paper can't review. I'm sure I've reviewed papers for conferences I've submitted to. Yes, there is the slight conflict that a negative review for some other paper will likely mean a higher chance of yours being accepted, but it's going to look mighty suspicious if the reviews for a paper are "accept", "accept", "accept", "strong reject" and the "strong reject" just happens to come from a competitor.

I think I would decline to review a paper that was on a topic very close to my own submission because a positive review might be seen as "Oh, he's just hyping the subject so we think his own paper's awesome, too" and a negative review as "Oh, he's just trying to kill off the competition." At the very least, I'd check that the PC member who asked me to review was aware that I'd submitted a paper on the same topic. And you can always check that the PC member is happy for you to review.

In conferences, even selective ones, the conflict of interest (as long as you do not review the paper of a past or current supervisor or supervised or obvious ally or enemy) is minimal. Also, good conferences concentrate the main experts on the field, so it would be very difficult to run it if they would perceive their reviews as conflict of interest.

This is quite different from reviewing for project proposals programs to which you submitted yourself. In these typically the total budget is very restrictive, and you are effectively placed in immediate antagonistic relation with all other submitters. Or else, if you end up being honest and recommend someone else's proposal in such a setting, if your own proposal ends up being rejected, you will keep asking yourself whether it was not your own recommendation of a competitor's that killed it. Or if you reject theirs, it is never clear how objective your judgement was.

And yes, as unbelievable as it sounds, it does occur that people are expressly and emphatically asked to review for calls to which they themselves submitted (e.g. because expertise is so scarce).

On the contrary, the IEEE conferences I am involved with (robotics and controls), explicitly expect authors to provide 2-3 reviews per paper submitted.

Reviews are not being submitted in a vacuum -- each paper gets multiple reviews, mediated by layers of editorial oversight. The downside risk of trying to game the system by submitting unwarranted bad reviews (losing the respect of senior members of your research community) far outweighs the minuscule potential benefits (the tiny chance that your paper is on the dividing edge of a hard quota line, such that bumping one other paper down bumps you over the line [if such a ranking/quota even exists], and that your bad review of an otherwise decent paper will tip the balance enough on it to bump it down when combined with the other reviews).

Assuming the papers are part of a blind peer-review process, I've never seen an example of this cited as a conflict of interest. Doesn't mean it isn't a potential conflict; I haven't come across this as such.

What would the conflict be in this regard? Are you worried about providing accurate feedback on papers? Usually, as a reviewer, there's a system in place whereby you're not just saying "yes" or "no" without giving written feedback.