Is the word 'places' an 'adverb of place'?

It depends of context and its, hm, placement in sentence)

You see, there is a word 'place', which can be noun in single form ('a place, the place'), noun in plural form ('places') and also it can be a verb. Verb, in its turn, can be used with nouns of first or second ('We place the new rug on the porch') and third person ('He places keys in his backpack') So to determine you just need to check - is a word 'place' a verb or a noun.

Let's take your example - "The dog takes his bone places'. Let analyze word by word:

• the dog - okay, it's our main subject, it is a noun word;
• takes - aha, its our subject's action, third person form, simple tense - so, it's 'take', done by third person - the dog;
• his bone places - well, looks like it's a subject of noun action, it can't be the verb - so it's a noun.

Looks like in current sentence context 'bone places' - it's a couple or multiple holes, done by dog for hiding its bones.