Why would a dryer receptacle trigger another receptacle to quit working
CALL YOUR UTILITY. NOW!
I'll bet you have either lost neutral or a lost leg. Either one can produce nasty results. My best guess is a lost leg - that would result in 1/2 your 120V circuits getting 120V, 1/2 your 120V circuits getting nothing at all unless a 240V appliance like a dryer is turned on and your dryer getting 120V instead of 240V (and therefore you get motor and controls but either no heat or minimal heat).
This can be an internal problem - e.g., loose wire in your main panel. But it is quite likely that it is a problem with the power getting from the street into your house. Call your utility. Tell them you think you have a lost leg or lost neutral. If they have no idea what you are talking about, just tell them you have an intermittent power outage and are concerned about safety.
In (almost all) places, they will check it for free and if the problem is outside they will fix it for free. If they determine it is a problem inside your house then they will tell you to call an electrician. If you start with an electrician then (unless it is a close friend or relative) you will most likely have to pay the electrician for a visit and still end up calling the utility to fix it. So start with the utility company.
Meanwhile, unplug any sensitive electronics or expensive appliances, as the problem may result in varying voltage which can really ruin things pretty quickly.