Is it better to use ice or heat after straining one's lower back?

Do what helps. Traditional wisdom is ice for first 24 hours to limit inflammation then heat after to reduce spasm and stiffness. While this approach works well for some injuries (like a sprained ankle) i don't find patients usually get much relief from the ice portion (likely less inflammation and more muscle spasm). Heat helps most people but in the end, do whatever works for you.
Ice. Ice can only help in terms of pain reduction and swelling reduction. It also numbs the skin nerves, so you feel better sooner.