There's no specific confirmation in any of the TV shows, films or official novelisations whether (or not) Romulans have a Katra. Their general lack of telepathic abilities would suggest that they don't have the ability to use them to transfer knowledge, but then again, who knows what goes on at the highest echelons of Romulan society or behind the closed doors of the Tal Shiar...?
We have a couple of Extended Universe sources that indicate that Romulans almost certainly don't have Katras, or at the very least if they do, then they're entirely unable to do anything useful with them;
In Starfleet Academy #15 - T'Priell Revealed, Part II of III we see a Romulan archaeologist and specialist in proto-Romulan cultures who doesn't know what a Katra is, nor how to use one. This seems proof-positive that Katras aren't a thing in Romulan society.
In TOS: The Lost Years we find that the teachings of the 'mind-lords' (instrumental in allowing the Vulcans to learn how to control and share Katras) never made it to Romulus when the early Romulans split from the Vulcans. It would appear that their telepathic powers are crude and ill-formed in comparison and that the level of mental sophistication required to share a Katra would be well beyond them.
“When the Romulans separated from the Vulcans,” Spock continued, “none
of the old High Masters went with them. Those on Vulcan, after Zakal’s
death, chose to align themselves with Surak. The discipline underwent
a radical change in philosophy. The path of Kolinahr still harnesses
great mental powers—but, for the sake of peace, applies them inward,
to the control of one’s mind and emotions, rather than using them to
control individuals and the outer environment.”
Star Trek: The Lost Years