Do Springer, IEEE, Elsevier charge a fee for non-open-access journals?

I agree with the other answers, but they are anecdotal, and you asked for some "definitive" answers to help you convince your colleagues and advisor. Here's what I found:

IEEE

None of the IEEE journals has a required charge for non-open access publications.

The publication FAQ says:

For a detailed listing of paper charges by publication, download the List of Voluntary and Overlength Paper Charges (PDF, 287 KB).

Here is the current version of that list. You will note that some journals have voluntary "sustaining" charges for "normal-length" manuscripts. In fact, IEEE policy specifically states that those charges are not obligatory:

IEEE Policy 6.9 permits some types of periodicals to levy page charges. If your publication is one of these, it is your company or institution, not you, which is being asked for support. Payment is not obligatory nor is it a prerequisite for publication.

In the statement above "page charges" refers only to charges for not-open, not-overlength, not-color-print articles. Many IEEE publications do have "overlength" page charges, which are mandatory - if your paper exceeds the "normal" length then you must either reduce the length of your paper or pay the overlength charges. The length of a "normal" paper depends on the journal and publication type (normal-length "letters" may be as short as 1 or 2 pages, for example.) Some also have charges for color figures appearing in print, although there is no charge for online-only color.

More definitive info is available for each journal in the IEEE Xplore site, at the "About Journal" page, section "Author Resources", link "Additional Information", which opens a PDF with detailed "Information for Authors" (see an example).

(The document also does not include the recently added IEEE open access options, which do have required charges, which are listed here.)

Springer

The Springer author FAQ asks

HOW MUCH DOES IT COST TO PUBLISH IN A SPRINGER JOURNAL?

The answer there is:

For the majority of Springer journals, publishing an article is free of charge.

• If a journal requires page charges, you will find them on the journal's springer.com homepage or in its Instructions for Authors.
• Charges might apply e.g. for color figures or over-length articles. The information on these extra costs will also be available from the journal's homepage or its Instructions for Authors.
• Charges for open access articles: If you choose to publish an open access article ("Springer Open Choice" article or in a "SpringerOpen" journal), there will be an Article Processing Charge (APC) to be paid by the author. As a consequence, the article can be freely read, downloaded or distributed from SpringerOpen.com and SpringerLink.com by any internet user.

Furthermore, even for open-access journals, Springer advises:

SpringerOpen journals routinely waive charges for authors from low-income countries; and individual waiver requests are considered on the grounds of hardship on a case-by-case basis.

Another Springer FAQ titled Why publish with Springer? claims:

Page charges do not exist.

which I think is as close to "definitive" as you can get.

Elsevier

Some Elsevier journals do have author charges, and by searching the Elsevier site I did find a (very) few non-open journals with required page charges. I did not find any in CS with author charges, though.

The official policy I found is:

I am submitting my paper to one of your journals. How do I find out if there are page charges associated with this journal?

Page charges are journal specific. Many journals do not have page charges for submitted papers.

To determine whether the journal to which you are submitting has page charges, you will need to find the homepage for the journal and check it's specific submission instructions.

So if you're looking for a definitive "No page charges," Elsevier is not the answer. In practice, however, there are literally only a handful of non-open Elsevier journals that charge author fees.

I have never paid for the publication of any journal paper, including papers in Springer, Elsevier, IEEE, ACM, SIAM, and AMS journals.

Some of those journals do advertise "page charges", but in my experience these are strictly voluntary. I've never paid them, and my papers were published anyway.

To be certain, you'd have to check the specific journal's author instructions.

However, I've published in two Elsevier and one Springer journal (in mathematics) and never paid an author fee. The Springer journal had an open access option, which would have had a fee, but we didn't choose that option.