No, use of such symbols is not required
You may not, without permission, use trademarks in such a way that a reasonable person would believe that the trademark holder has endorsed or sponsored or approved your product (or service) or that your product comes from the trademark holder or the same source as the trademark product.
Use of the ™ or the R-in-a-circle symbol can help make this clear. Better is a more explicit disclaimer, possibly using a symbol. Something such as
OtherProduct™ is a trademark of Acme Corp. Acme has not sponsored, endorsed or authorized MyProduct, nor is MyCompany in any way affiliated with Acme.
Including such a disclaimer without the symbol should also do the job.
It is permitted to say things like "MyProduct accepts files output by Acme's OtherProduct." or "MyProduct is 30% faster than OtherProduct." (provided that they are not clearly false statements. Use of the symbols in such statements may help make it clear that you are not claiming rights to those marks, but is in no way required.
The C-in-a-circle really has no use other than in copyright notices. No one that I know of even asks others to use it when referring to the name of a copyrighted work. The plain-text form "(C)" was never given effect under US law, although many used it. The forms specified in 17 USC 401 are C-in-a-circle, "copyright" and "copyr." But since 1989 all copyright notices are strictly optional under US law anyway (and in the law of most other countries also) using "(C)" will probably not cost the author any significant legal rights or protections.