The former England hooker played in Saturday's loss in Toulouse but felt ill earlier this week and was admitted yesterday.
He is undergoing treatment and further tests.
A club statement read: "The club’s medical staff and player welfare manager are closely supporting Shaun and his family during his recovery.
"The infection will rule the Robins skipper out of this Sunday’s clash with Widnes Vikings at KCOM Craven Park and next week’s Million Pound Game should the club feature in that match.
"Everybody connected to Hull KR wishes Shaun all the best for a speedy recovery and we look forward to welcoming him back for pre-season training in November."
Ex-Huddersfield Giants star Lunt, 31, has played just 16 games this term after a raft of injury issues but remains arguably the club's most influential player as they bid to avoid a potential relegation.
The Cumbrian started his professional career with Castleford Tigers in 2005 and, after making his mark with Workington, joined Huddersfield.
He won the Grand Final while on loan with Leeds Rhinos in 2012 and is still rated one of the best hookers in Super League.
Rovers host Widnes Vikings on Sunday knowing they will need victory in their final Qualifiers game to have any chance of guaranteeing a top-three place and securing their safety.