Wow, that's hilarious
I have the following suggestions when fighting shooting-heavy Dwarves with Elves:
1) Eagles. They're just great. If you can reach a warmachine, that's fantastic. If you can't, then your opponent probably wasted missile fire on them. If you have Eagles left alive when you reach the fight, use them to redirect those big Dwarf blocks. You don't want 40 great weapon warriors hitting you in the flank while you're already busy with another fight, and Eagles will prevent that. Eagles are great against every opponent, but they're generally game winners for me versus Dwarves.
2) Archers. If you have Archers in your list, you have some choices. If they have clear Line of Sight to enemy warmachines, consider just shooting at them. The Organ Gun in particular is a good choice; it can't take extra crew, so 3 Wounds and it's dead. A smart Dwarf player will keep his machines Obstructed or deployed in cover, but he may get careless. If you don't have clear shots to those warmachines, then consider advancing the Archers and using them like Eagles to redirect and get into the enemy backline.
3) Magic. Well, duh, magic is important. However, Pit of Shades, Dwellers and Purple Sun all insta-kill warmachines. Sometimes it's useful to pressure Dwarven magic defenses with these nuke spells early on, forcing him to burn his dispel scrolls, so that you can use magic more effectively when combat starts. If you even manage to kill a machine in the process, then that's a nice benefit.
4) Don't think you can avoid taking casualties, because you can't. I see a lot of players try to combat gunlines by sitting around in their deployment zone hoping to take out the warmachines first, or else following some long convoluted deployment plan that involves hiking a roundabout path around impassable terrain in the hopes that they can avoid being shot at. My advice for Dwarves is to send everyone into no man's land as directly as possible, weather the casualties, and focus more on dashing across the table as quickly as possible. If you're efficient with your deployment and movement, you'll overwhelm the Dwarf player's shooting by offering him too many targets to handle. It's a much better choice than going across the table peacemeal and giving the Dwarf some easy choices for target priority.