Yes Lemonade does eventually go bad if you do not keep it frozen or refrigerated.
You can keep lemonade for 6 months to a year if you freeze it and then thaw it out when you're ready to drink it.
If you keep lemonade in the refrigerator then it should last for up to a week or maybe a bit longer but if it taste bad then you should dump it out.
I buy lemonade from the store in plastic jugs on sale and then freeze them in my deep freezer which makes for some nice cold lemonade to drink throughout the year and especially during the summer.