Why can't the Dwarves bring ponies into Mirkwood in The Hobbit?

It was not that the Dwarves could not bring ponies into Mirkwood at all; instead, it was that they could not bring the ponies they were riding because those specific ponies belonged to Beorn, the man who could change his shape into a bear. Mirkwood had once been called Greenwood, but when the spirit of Sauron took it over and went to inhabit the fortress within it (Dol Guldur), it became dangerous and dark, and the Giant Spiders infested it (along with a lot of other things that were dark and dangerous in nature)—after that, it was known as Mirkwood. Beorn knew of the dangers and he had a close bond with all of the animals in his home (he had dogs, horses, and bees, among others), so he did not want his ponies to go into Mirkwood for fear of them never returning, which is why he told the Dwarves to set the ponies loose before they entered the forest.

In The Hobbit, why does Gandalf advise against bringing the ponies into Mirkwood?

Gandalf told the Dwarves to let the ponies loose before they entered the forest Mirkwood because Beorn requested that they do so. Beorn knew the dangers of Mirkwood: the Giant Spiders, the dark enchantments, and the fact that Sauron had taken residence in the fortress in Mirkwood, Dol Guldur. Because of those dangers, Beorn did not want to risk the lives of his ponies, whom he had a strong bond with (Beorn had a strong bond with all of the animals who lived with him, including the ponies, dogs, and also bees), so he made it quite clear that he did not want his ponies to go into the forest. Beorn also followed them to the forest to make sure that the Dwarves did as he asked, and then went home with the ponies once they were set loose.