Learn English – Should I write mobile phone or mobile phones

"Mobile phones have become an indispensable part of our lives."

The sentence as it is phrased is saying mobile phones as a group of things are only one of many parts or aspects of our lives, but they are a part that is indispensable. I might say "My pets are an important part of my life." so the fact that "lives" is plural isn't really relevant.

You could omit the prepositional phrase "of our lives" and the sentence would still be correct grammatically, even though it's not very enlightening. You could also replace it with a different prepositional phrase that uses a singular object, like "Mobile phones have become an indispensable part of our plan for world domination." The prepositional phrase is acting as an adjective describing part.

You would pluralize "parts" if you were talking about how something affects many different aspects of your life or the way you live, for example, "Almost all parts of my life have been changed by technology."

"Any" can be either singular or plural depending on the context.

"I will marry any girl who answers my email." There are many possible girls out there, but I am presumably only going to marry one of them.

"Any employees who work overtime will receive a bonus." Many employees might qualify.

In your example, the word "of" makes it so that it is not clear whether it is possible for many devices to be left on. That is, if you had said, "If any device ...", "device" is singular, so clearly a singular verb is called for. If you had said, "If any devices ...", then a plural verb would be called for.

The question then is, in context, is it only possible for one device to be left on, or could many devices be left on? I'd guess many, in which case the verb should be plural.