The hosts go into the final day on 199-6 in their second innings, a lead of 278 runs.
John Anderson top-scored with 59 while Ed Joyce (37), William Porterfield (29) and Gary Wilson (25) also contributed on a day when 45 overs were lost.
Tanveer Ahmed and Tanwir Afzal each picked up two wickets for the tourists.
The Irish play positively, scoring at a rate of around four an over in an attempt to force a result despite the best efforts of the Belfast weather.
Joyce scored a fluent 37 from 55 balls in an opening stand of 53 with skipper Porterfield and was unluckily dismissed when he hit his wicket.
Anderson and Porterfield added a further 48 before the captain was bowled.
Niall O'Brien (15) and Paul Stirling (six) both went cheaply, but Anderson's fourth half-century for Ireland ensured they were always on top.
The Merrion man top-scored before mistiming an attempted hook shot.
Gary Wilson scored a breezy 25 and was trapped lbw two balls before the umpires decided that the light was too poor to continue.
Porterfield indicated that Ireland would bat on Friday, probably hoping to set Hong Kong a target of around 330.
Play on the final day will start 30 minutes early at 10:30 BST, with a potential 104 overs.
Meanwhile, Ireland have brought North Down leg-spinner Jacob Mulder into their squad for two T20 games against Hong Kong at Bready next week.
He replaces Barry McCarthy, who is required by Warwickshire for their County Championship clash with Yorkshire at Headingley.
Ireland remain in control against Hong Kong in the Intercontinental Cup game but they were frustrated by rain and bad light at Stormont on Thursday.