Except for appellate briefs, service by fax is allowed in Illinois unless a lawyer has specifically said no. Here's how to say no--
Any party may rescind consent of service by facsimile transmission in a case by filing with the court and serving a notice on all parties or their attorneys who have filed appearances that facsimile service will not be accepted.
Illinois Supreme Court Rule 11(b)(4)(i). Keep in mind, however, that if you say no to receiving service by fax in a case, you aren't allowed to serve others by fax in the same case.
Is Rule 11 outdated? It was last amended in 1992. Meanwhile, service by email is becoming the norm in federal courts. While email has some problems of its own, it's certainly better than fax in document-intensive cases like mass torts, where parties might routinely file hundred of pages a week.