Direct proof of use
The BGM-71 TOW is documented in the 2014 War Against the Islamic State through both transfer reporting and battlefield-use reporting. Military.com reported in November 2015 that the United States was supplying Iraqi Security Forces with Raytheon BGM-71 TOW missiles for the counter-ISIS ground fight in Iraq.
Islamic State use is separately documented in Syria. Long War Journal reported that an Islamic State video from the Palmyra campaign showed at least one U.S.-made BGM-71 TOW being used against Syrian government troops near the city, and later reported Islamic State photographs from the Hasakah offensive that showed two instances of TOW missile use.
Sources: Anti-Tank Missiles Counter ISIS Ground Game, Long War Journal Palmyra TOW, Long War Journal Hasakah TOW
Timeline
The available open-source timeline places the strongest direct-use evidence in 2015. On June 9, 2015, Long War Journal reported TOW use in Islamic State's Palmyra campaign, based on video released by the group. On June 27, 2015, the same outlet reported photographs from the Hasakah offensive showing Islamic State use of U.S.-made BGM-71 TOW missiles.
By November 2015, U.S. supply to Iraqi Security Forces was publicly reported as part of the counter-ISIS ground fight. That source describes the TOW as a heavy anti-armor missile option for Iraqi partner forces, but it supports supply and fielding rather than a named Iraqi firing incident.
Sources: Long War Journal Palmyra TOW, Long War Journal Hasakah TOW, Anti-Tank Missiles Counter ISIS Ground Game
Operational role
For anti-ISIS coalition and partner forces, the sourced role was supply of anti-armor missiles to Iraqi Security Forces during the ground campaign against Islamic State. The Military.com report describes the missiles as BGM-71 TOWs made by Raytheon and frames the transfer around the counter-ISIS ground fight in Iraq.
For Islamic State, the sourced role was captured or otherwise obtained anti-tank guided missile use in Syria. Long War Journal attributed the Palmyra evidence to an Islamic State video and the Hasakah evidence to Islamic State photographs; it assessed that the missiles used at Palmyra and Hasakah were likely captured from battles with Free Syrian Army forces elsewhere in Syria.
The evidence therefore separates supply, possession, and use. The Iraqi Security Forces claim is a supply/fielding claim for partner forces, while the Palmyra and Hasakah items are direct battlefield-use claims attributed to Islamic State media and reported by Long War Journal.
Sources: Anti-Tank Missiles Counter ISIS Ground Game, Long War Journal Palmyra TOW, Long War Journal Hasakah TOW, TOW Weapon System