Active dislike of droids happened. The reasons for it varied.
Wuher (the Cantina bartender) had the following attitudes and reasons for them: (src: "Be Still My Heart - The Bartender’s Tale" by David Bischoff, from Tales From The Mos Eisley Cantina):
Still, as the human bartender bustled through the busy streets, sun hood up, squinting, he was bothered by that droid who had accosted him. Wuher was well aware that droids were essentially harmless. To hate them was like hating your latrine or stove or moisture vaporator if they’d somehow been overlaid with innocuous consciousness. True, droids tended to be essentially faithless, with no ethical or racial structure. So were a lot of biological aliens that Wuher had met. The truth, the bartender knew, was that droids were an easy target.
Wuher had been abandoned on Mos Eisley in early youth, a human amidst peoples who disliked humans. He’d been kicked about and spat upon all his squalid, hard life. His boss hated droids essentially because they didn’t drink and thus took up necessary room in the cantina that might be occupied by paying customers.
Wuher hated everyone, but droids were the only creatures he could actually kick with impunity.
So, in the case of the barkeep who told Luke to get his two droids out, it wasn’t so much "racism", as overall hatefulness. Droids weren’t really hated for being droids - they were hated same as anyone but were the only ones to whom the hate was expressed openly due to being the only safe outlet for it.
Revelation - at the end of the story, the bartender hires the droid, who ends up being a super-tremendous help (by making a new drink that Jabba the Hutt likes… using the shot-first-Greedo as an ingredient). Bon Appetit!
In the case of the Wookiee Cantina owner, it was pure pragmatism (he would likely have hated any species who didn’t eat or drink at his cantina as a species).
In the films, we also see Han Solo exhibiting less-than-positive attitude about C3PO, but that was mostly just Han’s abrasive scruffy personality… he wasn’t exactly much more pleasant to many if not most biologicals.
Another example in the EU was Marvel old time comics.
Those had Valance. Quoting from http://www.hftf.co.uk/?tag=droid-racism:
Valance’s motivation is a little shaky to say the least. He hates droids. Like, really hates them. He hates droids so much that he’s made it his mission to hunt down Luke Skywalker, not because of any sizable Imperial bounty on the boy’s head, but because he’s heard a rumour that Luke actually likes droids. Surely you could find better things to do with your time. And who wouldn’t like Threepio and Artoo? They’re adorable!
Marvel’s Star Wars series really ran with that “people hate droids” idea in its early days. Just because that bartender in the cantina wouldn’t serve them we got a few stories centred around the concept. In the very first post movie storyline Han and Chewie catch a ton of shit for helping an insectoid priest take a dead cyborg pilot to a burial ground. Yeah people hate cyborgs too, which is a bit like hating people who wear contact lenses, but whatever. I’m really surprised that no bright spark has retconned this droid racism as being a hangover from The Clone Wars. Plenty of planets were invaded by the Trade Federation and the armies of the Seperatists, that it stands to reason that some people who remembered back twenty years would be a bit ticked off when they saw any robots. Job done. You can have that one, Lucasfilm.
So anyway Valance fails to find Luke – he runs into our old mate Jaxxon instead. Then we learn that, shock horror, Valance is a cyborg himself. In fact his major passtime, aside from blasting the shit out of innocent robots, is sitting in his cockpit, stewing in anger and self hatred, and ripping away his fake skin to reveal his true semi mechanical nature. It’s a well used image, that probably appeared in a million 50s comics and pulp sci fi novels, but it looks good, hence the fact they reused the scene a couple of times.