I am a remote worker — can I legally live in Israel on a tourist visa?

No. Israeli law prohibits performing work in Israel without a valid work permit, regardless of who your employer is or where your salary is paid. Working remotely from Israel for a foreign company is still "working in Israel" in the legal sense. The risk of enforcement has historically been low for short visits, but people who enter Israel repeatedly and stay for extended periods while working remotely face a growing risk of denial of entry or deportation. If you want to work in Israel long-term, explore the B-1 work permit or other legal channels.