Lobsters are not toxic after death. However, they carry bacteria that will multiply after death, and they can produce toxins, so it’s important to keep lobster below 40F or over 140F to inhibit bacteria growth (same with most other protein sources).
Like fish, they are super-delicate, especially organs and gills, and those body parts will decompose very quickly, so removing the meat from the body as close to death as possible and refrigerating/freezing the meat is usually best if you aren’t immediately cooking it.
