What verbs put the speaker in the Dative case?
Quite a lot of them, actually. For a list, see the link in the footnote of this answer. It's not small enough to be copied here...
In my book, this is called "дательный восприятия", that is the "dative of experiencing". It's used when the subject is experiencing some condition or sensation. That may explain why many such verbs are reflexive (ending in "-ся" / "-сь").
- "Мне нравится Достоевский" = "I like Dostoevsky"
- "Мне запомнился Париж" = "I remember Paris" (connotation: Paris struck me!)
- "Мне надоело ждать" = "I grow tired of waiting" (a few non-reflexive verbs work like this, too)
As already mentioned, experiencing doesn't necessarily require a verb. It can be expressed with any kind of predicate — adjective, noun, or pronoun:
- "Мне холодно" = "I feel cold"
- "Мне [вас] не слышно" = "I can't hear [you]"
- "Мне негде жить" = "I don't have any place to live"
- "Мне знаком этот голос" = "I know this voice" (connotation: It sounds familiar to me, I've heard it before)
- "Мне конец" = "I'm done for" (tragic connotation: It's all over for me, kiss me goodbye)
The dative is required with certain predicative idioms:
"Мне не до шуток", "Мне без разницы", "Мне и дела нет", "Тебе крышка".
The dative is required when talking about someone's age (no verb here, though):
- "Мне за тридцать" = "I'm over thirty"
- "Мне пятый десяток" = "I'm in my forties"
The dative is required if the predicate in a full sentence (most often a negative one) is changed from a personal form to the infinitive:
- "Я не оправлюсь от удара" = "Мне не оправиться от удара"
- "Я не стану президентом" = "Мне не стать президентом"
- "На чём же я езжу?" = "На чём же мне ездить?" (Vague connotation: If not this car, what else am I supposed to drive?)
In some cases it's possible to find an equivalent predicate and avoid using the dative (there might be a slight difference in style and connotation, though):
- "I need you" = "Ты мне нужен" = "Я нуждаюсь в тебе".
- "I can't hear you" = "Мне вас не слышно" = "Я вас не слышу".
A detailed analysys in Russian (it may be difficult to read, but there are good lists of verbs and predicates there): Корпусная грамматика. Дательный падеж.