Really? I've never gotten that. I know they're not close, and there's been stuff going down to tear them apart (Donna's death being the obvious), but I've never felt like they loathe each other. Especially not Tig loathing Jax
Tig has said that he doesn't trust Jax (meaning, he doesn't trust him to understand when death and mayhem are required and to have the guts to go through with it); there were a few explicit mentions of this in season 1 (for instance, Tig said exactly this to Clay in the very first episode, and then he said it again early on in the second season). The essence of their problem is that Tig believes that violence is always the answer and he doesn't reflect or consider alternatives, he wants immediate action. Jax always tries first to find a non-violent solution if he can. Thus the mutual distrust.
And for expressions of loathing, two come immediately to mind. The first was back in season one, in the episode when they were scrambling to find the $200K for the IRA: Jax swaggered into the workshop with the $15K they needed, having sold Oswald's diesel to Uncer - the look of pure loathing on Tig's face was absolutely classic. That followed on from an earlier scene where Tig had wanted to go and do violence on someone for selling crank but Jax had proposed that they go and ask the truck driver where he bought it (in other words, he was looking for evidence before acting, whereas Tig just assumed the cause and wanted immediate action) - when Clay gave Jax permission to go question the driver, Jax yanked his cut out from under Tig (who was leaning on it) with emphasis, and Tig gave him a really filthy look. Another example where this loathing is really clear is in the episode when Jax stops Tig from killing the girl witness in the safehouse. There have been a lot of those little incidents: there is some real tension between those two, and it has not gone away.
On the other hand, in Home (3.04), they hug and say they are glad to see each other when Jax and Clay turn up at Gemma's dad's home, and at the end of Widening Gyre Jax seems very sincere when he says that they need Tig with them in Ireland: I have taken these signs not so much as evidence that the problem has disappeared but as a symptom of Jax's inner demons re-prioritising. If they ever get out of survival mode, those issues (just like Donna's death) are going to come back to the fore.
ETA: this is off topic but I have always wondered about it and there was no one to ask at the time: Jax sold the diesel to Uncer for $20K but it was clear that he only gave $15K to Clay - do you think he kept the extra $5K for himself? It doesn't seem like a very Jax thing to do (I never get the sense that money is very important to him) but the anomaly is clear.