There has been nothing stated about Eobard Thawne's sexuality in-universe or out of universe at this time. The character is all about ambiguous actions, playing his cards very close to the chest, for obvious villainous reasons. His love/hate reactions to Barry being the most blatant (he "respects" Barry but you know, still wants to kill him). If anything, his actions show that he is a pragmatist and very calculating. He's playing a long game in order to kill Barry while still being able to get back to his own time. He's likely just playing it up as needed, like pretending to be paralyzed, until it's no longer useful. We know that he's been ready to get out of the chair when his life is at risk (when the robotic bees swarm the headquarters).
Then again, he is from the 22nd century (25th in most comics), who knows what sexual identities exist five generations from now (2155).
Of course, comic Hartley, in the New 52 reboot from 2011ish, was tied to a relationship with Captain David Singh, who is confirmed Gay, and eventually married to Rob, in the series. The Draft script of the Pilot has Hartley still with Team Flash, stating that he, not Cisco, designed the firefighter suit, while dating a fireman. So the ho-yay feels might come from Hartley. Coincidently, the actor Andy Mientus is also openly Bisexual.