The short answer to the question that forms this thread's title is: "no".
Unpacking that, the key words in the second half of the verse, Ἀσπάζονται ὑμᾶς οἱ ἀπὸ τῆς Ἰταλίας, simply refer to native Italians in whatever place the writer was currently located.
This is a long-held view: see for example Marcus Dods' explanation in The Expositor's Greek Testament (1897), vol. 4, p. 380-381 (basically, just the explanatory bit on p. 381, in fact). This same view is represented in the somewhat more recent Eerdmans Dictionary of the Bible, gen. ed. David Noel Freedman (Eerdmans, 2000) -- see the article "Hebrews, Epistle to the" by James W. Thompson (pp. 568-570). He appeals to the parallel in Acts 18:2, and goes on: "This parallel suggests, therefore, that the author of Hebrews writes from a location in which he has met expatriates 'from Italy' who send greetings to their home city" (p. 569b).
Unfortunately, the inherent ambiguity precludes using this as evidence for provenance of author or recipients.
As for the further question:
What evidence is there as to the letter's place of origin?
Well ... there isn't any! All suggestions remain speculative.