The Spanish team looked to have booked their place in the last four of the competition but late strikes from Marco Reus and Felipe Santana condemned them to a 3-2 aggregate defeat in their quarter-final tie.
It's not clear who Al Thani was directing his accusations at, however, replays suggested that Santana's goal should not have stood as he was in an offside position.
"I'm sorry to go out this way injustice and racism," Al Thani wrote on his Twitter page.
"I hope to open a thorough investigation UEFA regarding the Spanish club out this way. This is not football, but racism."
Dortmund along with Real Madrid sealed their place in the semi-finals of the Champions League on Tuesday.