What one associates with "burning in the sun" is usually sunburn. Plants are very different from animals; they do not even have all of the same cell organelles that animals do. Taking this into consideration, plants do not have the kind of epidermal tissue (skin) that animals do, making it impossible for them to get sunburned.
What can happen to a plant from getting too much sunlight is dry out. Plants need water for photolysis, which is during the light-dependent reactions (when the sun is out) of photosynthesis. Without enough water and too much sunlight, a plant can catch aflame.
To prevent this, a plant's epidermal tissue secretes a waxy cuticle to prevent from dessication (drying out). Plants absorb the energy from the sun and use it for photosynthesis, which allows them to soak up water and prevents them from heating up as well. This is due to water's high specific heat, meaning that it takes a lot of energy to heat water. They make ATP (usable energy) when the sun is out, and use that ATP later to synthesize glucose when the sun isn't out (the light-independent reactions). ...MORE
while mostly true the former answer misses one major point. plant do get sun burned and not only dry out only they don't get burned the same way we do. sunlight contains many light wave lengths, some are high energy like UV which can be harmful to all living cells as it damages DNA. another problem with the sun is that in the process of absorbing the energy from the sun and harnessing it the process sometimes "overflows" a bit like a nuclear reactor overheating which creates free radicals. these are highly active molecules which can wreck havoc on living cells mainly by ruining membranes. plant produce many molecules that absorb and neutralize these radicals called antioxidants which are very healthy for us 'cos they help us deal with free radicals produced while breathing. most weedkillers do not directly harm the plant but instead damage these defense mechanism leaving the plant exposed to the harmful effects of the sun.