Answering for gen 6 because that's what this question was originally asking about
Physical Walls: Hippowdon, Skarmory, Chesnaught, Mega Venusaur, Amoonguss, and other physically bulky Pokemon have little trouble taking Lopunny out in most cases. However, they can struggle against sets that carry Encore and can be overwhelmed by Power-Up Punch. Physically defensive Gliscor and Landorus-T can also deal with it, but both fear Ice Punch. Notably, Cofagrigus can easily take an initial hit, remove Lopunny's ability thanks to Mummy, and completely wall it thereafter.
Fairy-types: Physically defensive Fairy-types, such as Clefable and Mega Altaria, have little to no trouble handling Lopunny, as they resist High Jump Kick and can take a Return quite well. Clefable in particular can run Unaware to ignore Power-Up Punch's Attack boosts or Magic Guard to ignore damage from entry hazards.
Bulky Psychic-types: Physically defensive Psychic-types can take any of Lopunny's STAB moves as long as they have a decent amount of Defense investment. Celebi is a very good answer to Lopunny, though it can be taken advantage of thanks to the combination of Encore and Pursuit Bisharp. Slowbro and Mega Slowbro take hits from Lopunny very easily and deal a lot of damage to it with Psyshock, and the former doesn't even fear Encore sets much because of Regenerator. Physically defensive Cresselia completely walls Lopunny and can hit back quite hard with Psychic.
Choice Scarfers and Faster Attackers: Choice Scarf users such as Landorus-T, Keldeo, Latios, and Garchomp as well as faster Pokemon such as Mega Aerodactyl, Mega Alakazam, and Mega Sceptile can outspeed Lopunny and take it out without too much difficulty. However, all of these get worn down by Fake Out and cannot always reliably switch in.
Priority Users: Talonflame, Mega Pinsir, Breloom, and other priority users can KO Lopunny before it gets to attack; however, much like faster Pokemon, they can't always switch in very well.
Residual Damage: Lopunny can quite easily be forced to take residual damage to the point where it is taken out. Iron Barbs, Rough Skin, and Rocky Helmet users such as Ferrothorn and Garchomp can take advantage of Fake Out abuse, though the former must switch out immediately afterwards for fear of High Jump Kick. Pokemon that carry Protect can use it to force Lopunny to take recoil damage if it uses High Jump Kick, but this can backfire if Lopunny runs Substitute, so it should only be used as a last-ditch effort. Sandstorm damage and Spikes can also chip away at Lopunny's health to some extent.
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