They do say... "some like it hot"... but there are limits!
I've visited the Canary Islands between the months of October and May, but I really do suffer when it gets too hot, so my preferred months are now limited to December to February. Of course, I only ever go there to walk, not to lie on beaches, and I generally head for the highest, wildest places, and keep as far away from the resorts as possible. Fortunately, La Palma has plenty of high and wild places, and hardly any resorts.
I have seen plenty of discarded tissues in some places, but from what you say, it might be worse in the summer months. From my experience, 'tissue trails' tend to be the most popular trails, presumably followed by people with limited capacities for 'holding it in'. Caves alongside some trails stink of pee, and it's usually easy to determine if that's because of people using them as toilets, or because of goats using them for shelter.
In the Caldera de Taburiente, which is a national park in a massive hole in the middle of the island, they are very strict about camping and littering. You have to apply for a permit to camp, and although it's free, numbers are limited, and you only get access to toilets, water and a visitor centre full of interesting information. One of the rangers was particularly keen to share his thoughts with me, just before a busy week was expected. He said that a lot of Germans and Spanish people camped in the Caldera. He said that he didn't particularly like the Germans, but admired the fact that they always cleaned up and took all their rubbish away with them. He always enjoyed great times chatting to the Spanish, but was dismayed when they often left the campsite strewn with rubbish, and continued to discard litter as they walked out of the Caldera.
Incidentally... anyone thinking of visiting the Caldera and camping there needs to be careful what food they take with them. Stoves of all types are BANNED. One spark would set the whole national park ablaze, and this has happened in the past. So... choose ready-to-eat food... and try and keep well away from the rangers base when they're doing their own cooking... because whatever they're having smells delicious!