First I'll play dumb.
According to this, a torrent file should have around 1000-1500 pieces. But let's take an especially poorly made torrent; a 4GB file (FAT32's maximum allocateable file size) split into 16kB pieces; that is ~250 000 pieces, so 250 000 write cycles. Common SD cards (and phone internal memories too) endure approximately 100 000 write cycles; so if you calculate this, you'll reach the conclusion that your card will fail before half of this torrent downloads. But this reasoning is wrong.
Each cell in flash memory endures 100 000 write cycles, and different chunks get written to different cells (overwriting the previous chunk with the next would be kind of dumb) - so even by downloading this one torrent, you will only use up 1 write cycle of this 4 GB area. So what counts is not the total number of writes, but the number of times a single cell has been written.
(Also, it can be noted that on most storage devices the computer does not write/read arbitrary lengths of data; it reads/writes a single 512-byte block at a time. So even if you write this file in one single pass, it will get written to the card divided into 512-byte blocks - so that means 500 million writes, but it's spread over 500 million blocks, so the wear is still one write cycle.)
Another thing to take into consideration is that SD cards and most flash controllers often do wear leveling, which means if the controller notices one block to be worn more than another, it swaps them internally to wear them equally, without the host system noticing. Also they include spare blocks to replace worn-out blocks - so even if you single out a block and try to wear it out, you'll find it will endure a lot more write cycles than it should, because halfway into the assault the controller remaps it. The whole card fails only when it runs out of spare blocks.