To reply to this question, we need to head back to the early years (the First Age of the Sun, that is), as so many times happens...

There is a grudge (and what a grudge) between Dwarves and Elves, yep. Or, better say, between some Dwarves and some Elves -- the Sindar. I won't spam you with a lot of quotes and other reflections but the main ones:

Dwarves sacked Doriath's treasury and killed Thingol (because all the tale related to the Nauglamír, the necklace the Dwarves made for Finrod Felagund). That's enough cause for Elves (Sindar) to, hrm, don't like Dwarves (and I must to say Thingol deserved it :P). Aside this, the other main cause would be that they woke up the Balrog of Moria, Durin's Bane). Besides, Dwarves are Aulë's children, not Ilúvatar's, as are Elves and Men, so they eye each other with some... suspicion. Elves are tall, fain, handsome; Dwarves are the contrary. Naugrim the Sindar named them, 'The Stunted People'. Would you like to be named thus?

The only Sinda friend to the Dwarves and to earn their trust was Eöl (and Maeglin his son), but that was before Dwarves killed Thingol. Oddly, Noldor (Fëanorians) and Dwarves got along very well (perhaps it's not that odd! ). Caranthir was their main source to trade; Curufin became their friend and learned Khuzdul and all. Maedhros saved Azaghal's life and the King of Belegost became one of his most trusted allies (aside Fingon :P).

Dwarves (Azaghal) gave Maedhros the famous Dragon Helm of Dor-Lómin as a gift (and Maedhros gave it to Fingon and from Fingolfin's son it passed to the House of Hador). Telchar forged Angrist for Curufin, that knife that Beren used to cut the Silmaril from Morgoth's crown.

Thranduil wasn't especially kind to the Dwarves; the Elves of Mirkwood fought the Battle of the Five Armies, yep, but also the Men of Laketown, all of them against the Orcs and Trolls. As for the people of Lórien, of course they didn't like Dwarves. Even if most of them are Silvan, their Lord was Celeborn, nephew to Thingol.

The same Celeborn that stayed in Ost-in-Edhil and didn't move to Lothlorien when Galadriel was kicked off (ahem) as ruler of the city. Why? Because he didn't want to enter Khazad-Dűm. Galadriel did, though, as she showed to be friend to Gimli -- but she was half-noldo too.

Ah! And before to end this post, don't forget Narvi, one of Celebrimbor's best friends. It is said there was a large and good frienship between Ost-in-Edhil and Khazad-Dűm -- but, again, Celebrimbor was a Noldo, a High Elf, grandson of Fëanor. There you have the Gates of Moria...

At the top, as high as Gandalf could reach, was an arch of interlacing letters in an Elvish character. Below, though the threads were in places blurred or broken, the outline could be seen of an anvil and a hammer surmounted by a crown with seven stars. Beneath these again were two trees, each bearing crescent moons. More clearly than all else there shone forth in the middle of the door a single star with many rays.
'There are the emblems of Durin!' cried Gimli.
'And there is the Tree of the High Elves!' said Legolas.
'And the Star of the House of Fëanor,' said Gandalf. 'They are wrought of ithildin that mirrors only starlight and moonlight, and sleeps until it is touched by one who speaks words now long forgotten in Middle-earth. It is long since I heard them, and I thought deeply before I could recall them to my mind.'
'What does the writing say?' asked Frodo, who was trying to decipher the inscription on the arch. 'I thought I knew the elf-letters but I cannot read these.'
'The words are in the elven-tongue of the West of Middle-earth in the Elder Days,' answered Gandalf. 'But they do not say anything of importance to us. They say only: The Doors of Durin, Lord of Moria. Speak, friend, and enter. And underneath small and faint is written: I, Narvi, made them. Celebrimbor of Hollin drew these signs.'

Nolen