I don't have any information about what case to use with 'plus' (or 'magis'). In dictionaries usually only prepositions take some case, and it is showed in parentheses.
In my language, 'more' takes the genitive case. What does Latin do?
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As Cerberus points out, plus is an adjective and has therefore the gender and number and case of the main word. There is also the corresponding adverb plus, which could be seen as the neuter accusative of the adjective. The word magis is also an adverb, but not synonymous with plus. I assume your question concerns the adverbial usage, but do bear in mind that the adjective one is possible as well. If you want to learn about the differences of the two uses, please ask a separate new question.
The adverb plus means "more" or "in greater amount" (but not "to greater extent"). If you want to specify what you have more of, you use the genitive. For example, "more money" would be plus pecuniae. The partitive genitive has many uses, so the use with plus should not be seen as a special case.
An adverb can behave much like a preposition, but the important difference is that an adverb can be used alone. You can simply cry plus! when you are hungry and want more food, just like you could demand "more!" in English.
The adjective plus means "more". If you want to say more than, you can either use the (often elliptical) conjunction quam, or an ablative of comparison.
[Ego] habeo plura capita quam homines [capita habent].
[Ego] habeo plura capita quam [ego habeo] caudas.
After quam, you would use the same case as the first element of the comparison, so ego and homines are both nominatives in the first sentence ("I have more heads than humans do"); in the second sentence, both plura capita and caudas are accusatives ("I have more heads than tails").
Habeo plura capita hominibus.
You can also use a simple complement in the ablative. In that case, it is not made explicit what the ablative is compared with, but that should be determined by context. This is mostly used when the first element of the comparison is in the nominative.
The usage of plus and other forms like plures, plura seems to be a little complicated and depend on the grammatical number.
The form plus looks like a singular neuter adjective in the nominative/accusative case.
However, what I've read is that this word in the singular is only used as an adverb (the nominative/accusative singular form of an adjective can often be used as an adverb in Latin), or as a noun (or "substantive").
If I understand correctly, the singular form plus wasn't used as an attributive adjective along with a neuter noun in the nominative/accusative singular. And plus does not even have attested masculine or feminine forms in the singular, which seems to rule out the possibility of it being used to modify a masculine or feminine singular noun.
As Joonas Ilmavirta said, the form plus can be used with a partitive genitive.
It seems that the partitive genitive could be singular or in some cases plural, even though plural nouns could instead be used in an alternative construction (discussed in the next section).
The Lewis and Short entry for multus (which plus is analyzed as the comparative of) gives the following examples (among others):
Singular genitive: pecuniae plus, plus honoris, plus auctoritatis
Plural genitive: plus virium, plus hostium
In the plural, plus apparently has not only the form but also the function of an adjective, with distinct masculine/feminine and neuter forms that would agree in gender and case with the plural noun that is being modified.