Well, let me look up the starting spellbook for the Beastmaster, having never played with it. While I don't like the distribution of cards, and it doesn't feel like it has enough of a coherent plan, you still have a decent number of tools in that build. You may still be able to hold off a Lord of Fire rush using a combination of Elemental Cloak, Lair spamming Timber Wolves (and possibly Emerald Tegu), stacking armor, Cobra Reflexes, and Block, but it's going to be somewhat precarious.
Guards can actually help a lot, although that works better in combination with Agony so that the Sweeping attack is much less attractive. The best guard options are Feral Bobcat, Timber Wolf, and Emerald Tegu.
Remember that you can put out Burns with Geyser if you get a bunch stacked on your mage.
In addition, due to the rules for multiple attacks and Flying, the Lord of Fire cannot use his Sweeping attack on a combination of Flying and non-Flying creatures (see the rules supplement). As such, you should probably play one Thunderdrift Falcon at a time. If your opponent kills it, replace it. You can put Pet on Thunderdrift Falcon, Timber Wolf, or Emerald Tegu. You can use the Falcon to hinder the Lord of Fire, which can allow you to run away from it even if it has Cheetah Speed. Just make sure to prevent it from having both Cheetah Speed and Mongoose Agility at the same time. Your opponent can cast Maim Wings on the Thunderdrift Falcon, but that's not very efficient, and you can play another one.
Mage Staff, Dragonscale Hauberk, and Agony will help.
Other options include Maim Wings, Force Hold, Enfeeble, and Turn to Stone. I don't think they're the best way to deal with a Lord of Fire for a Beastmaster, but there are quite a few control cards that will force your opponent to respond.
Tanglevine, as you noted, is a horrible solution to Adramelech. Use that on the Warlock (or a Dark Pact Slayer Blood Reaper if he plays one), not the Lord of Fire.
You can opt for the Gorgon Archer solution proposed by Ring, but I don't feel that it's necessary.
You should be playing either a Battle Forge or the Lair in order to get an action advantage.
There are plenty of additional options in the expansions that can be helpful, but there are quite a few tools available to a Beastmaster against an early Adramalech rush in the core set as well.