As always there must be agreement of the article in gender and in number with the gender and number of the noun; since "boutique" is féminine and in the singular and since the feminine singular article is "la" you should write "La petite boutique". The same principle is applied to adjectives (petite).
If "The Little Shop" is the title of a book or the name of a shop, you keep the upper case letters for the first letter of the title (La petite boutique) but you write the whole title in italics.
Independently of this principle of translation, if you want to name as shop in French you can use lower case letters if you prefer; there is no rule for that: the choice is considered to be a matter of style, of personnal taste.