Not necessarily. It could respect the car's wishes to sleep and stay asleep. This is how VT works even if you keep it running 24/7. If the car's asleep and you open VT, there's a little sleeping Tesla picture and a button where you can force it to wake up.
The bigger concern, IMO, is the security of the web-based sites. Depending on how they're implemented, you're at a fair bit of risk because not only can you unlock the car with your login, you can also find it, start it and drive away. So you have to have a fair bit of trust that both they've implemented things how they said they did, and that they've implemented them safely. For me it's a big ask to trust a $100k vehicle to a website.