When I first heard this question (in MQL, January 2018) I could find surprisingly little about it on the Internet.
Wikipedia doesn't mention cheese on its Italian Open (golf) page. Googling "italian open golf cheese" finds such snippets as "Steve Webster of England was presented with his weight in cheese after winning the 2005 Italian Open Golf at The Castello Di Tolcinasco Golf Club in Milan" (on breakingnews.ie) and "There is a "win–your–weight–in–cheese" bonus for taking the Italian Open title here at Gardagolf tomorrow [4 May 2003] and it could add up to an awful lot of parmigiano if the halfway leaderboard is any guide" (on the Irish Times). The Times's point is that the Australian Peter O'Malley ("tipping the scales at over 14st") had a share of the lead, and just three strokes behind was none other than Colin Montgomerie.
The actual winner of the Irish Open in 2003 was Mathias Grönberg of Sweden, who (according to Wikipedia) is no lightweight at 86 kilograms (190 lbs, or 13 stone 8 lbs).
The Daily Telegraph also mentions Steve Webster's 2005 win; apparently he weighed 75 kilograms. The Daily Express, in 2017, reported that Graeme McDowell won 80 lbs of Parmesan in 2004. Surely not his own bodyweight, but ...
That's three consecutive years covered: 2003, 2004 and 2005. I haven't been able to find out how much longer, if at all, this 'tradition' actually persisted for. I hoped there might be a clue in the tournament sponsors, but no; from 2002 to 2007 the Italian Open was sponsored by Telecom Italia. Nothing to do with cheese, as far as I know ...
© Haydn Thompson 2018