No, because your body tries to protect you from that outcome.
If your DNA get's damaged by UV light, it has an built-in repair mechanism (nucleotide excision repair). If the UV(B) dose is too high for the repair mechanism to work, there are other ways to protect yourself. An inflammatory response happens, which causes the skin to redden, hurt and itch (NSAID like aspirin or ibuprofen help with the pain). This is what you see when you have a sunburn. Your cells also undergo apoptosis, as Erastor mentioned, which is why the skin can peel off when you have a sunburn.
Sunburns are not the cause of skin-cancer. The DNA damage that's caused by UV(B) light that
cumulates over time is. But since a sunburn is a sign of heavy DNA damage, it is an indicator for this. Cumulative UV(B) damage induces mutations that break the system the body uses to protect yourself from cancer. This leads to uncontrolled cell proliferation, leading to cancer. Unprotected exposure to dangerous doses of UV(B) light (and sunburns as a sign of this) increase
the risk to get skin-cancer the longer the exposure over a lifetime.
The fact that you've had pre-cancerous lesions removed show that some of your cells already did get damaged too much over time resulting in, most likely, an
actinic keratosis.
It's very good that you seem to get a checkup and that the pre-cancerous areas have been found and removed! That is very important, especially in sun prone
regions. That and using sun screen (UV A
and B!).