George takes the dead mouse away from Lennie and says, "You ain't petting no mice while you walk with me." At first, then, we think that George disapproves of Lennie's petting mice. After Lennie retrieves the mouse, however, and George throws it away again, Lennie breaks down and cries. George explains, "I ain't takin' it away jus' for meanness. That mouse ain't fresh...and you've broke it pettin' it. You get another mouse that's fresh and I'll let you kept it a little while."
George understands that Lennie likes to touch soft things, which are a comfort somehow to Lennie; however, because Lennie doesn't know his own strength, he typically kills small animals. That the mouse is dead is what disturbs George, I think, because it could carry disease and make Lennie sick. George is willing for Lennie to have another mouse so he doesn't disapprove of Lennie's having mice.
In Chapter One of Of Mice and Men, George discovers that Lennie has a dead mouse in his pocket and he reacts with anger. He demands that Lennie hands it over ("Give it here!") and cannot understand why he would want to keep a dead mouse in his pocket. Moreover, George gives Lennie a warning against petting any more mice, dead or alive:
"Well, you ain't petting no mice while you walk with me."
This incident with the mouse reveals much about George's relationship with Lennie. He is far more than a friend or traveling companion to Lennie. In fact, he is almost like a parent since his primary concern is for Lennie's welfare. Note how he scolded Lennie in the earlier paragraph for drinking too much water. George is, therefore, responsible for Lennie and knows that he must keep him on the right path if they are ever to realize their dream.
I believe George doesn't hate Lennie all he is doing is protecting him. George already knows how Lennie is.
I believe that George don't mind Lennie petting the mice as long as its not dead when its dead he don't like it at all.