With just two minutes left on the clock, Wight had a relatively straightforward chance to win it for the Warriors but he was off target, ensuring Glasgow failed to follow up their historic win over Leinster back on home turf.
"It's incredibly disappointing to have lost," said Lineen. "The crowd wanted us to win so much.
"It's not that the effort isn't there or that the players don't want it.
"There was a lot of missed opportunities tonight and we didn't handle Treviso's game plan well. I would have to ask our players if a lot of the decisions they made they would make again. However, Burton controlled the game brilliantly from ten and took his chances.
"We must just keep learning, but the lows are very low and we're not feeling good right now. We really wanted to deliver a good result for our fans, and it just didn't happen."
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