I am not a native speaker but I will try to answer. Consider the following ways to express obligation:

Il nous faut partir.

Nous devons partir.

Il faut que nous partions.

The 2 and 3 are equivalent. The 1 is equivalent for the sense, but more stylish if not literary (but grammatically perfect). Some natives find it no colloquial French at all and even consider it snobby.

See the question here.

Therefore for your particular case:

Il faut que tu fasses la vaisselle=Tu dois faire la vaisselle.

On the contrary

Il te faut faire la vaisselle.

is not considered colloquial French (despite being grammatically impeccable).

In this construction, « il te faut » is colloquial, barely acceptable ; on top of communicating the usual idea of obligation in “il faut que “, it is meant to attenuate somewhat the idea of slight forebidingness and oppressiveness that go with telling people their obligations. Nevertheless, it might be asked whether this construction which is standard for the meaning of « falloir » as « avoir » is not issued from this usage.

• Il te faut une carte annuelle pour ton bus scolaire. (normal speech, not colloquial)
• Il te faut des lunettes de soleil lorsque tu est dehors, les ultraviolets détruisent la vue.
• Il te faut un lave-vaisselle.

No, in the first sentence the mode « indicatif » is used only for « falloir » (faut) ; then the mode used is the infinitive (faire), and that mode operates as the subjonctive mode : the action is being refered to generically, not as taking place or having been carried out, and so on.

The two sentences are used to speak generally ; the fact that « te » is used is a matter of register ; as said, the register is quite colloquial ; when used “globally” the idea of attenuation has no reality. The use of the second person singular when expressing what people do is equivalent to the use of the second person plural, just as with the use of « you » in English ;

• Le samedi et le dimanche tu ne travail pas dans leur pays, ici tu travaille même le dimanche matin.
  In their country you don’t work on saturdays and sundays, here you work even on sunday morning.

• Dans ce camp de vacances il te faut faire la vaisselle et les lits, mais le prix que tu paye pour le mois est plus intéressant.
  In this summer camp you have to do the dishes and make your bed, but the price per month is cheaper.

The Following ngram shows “il te faut” is not used as much as “il faut que tu” ; in particular it is not used with “faire” and we find instead “il faut que tu fasses”.

Il faut que tu fasses la vaisselle is the standard way to say to someone that he needs to do the dishes. It can be either a direct order or strong advice (do it now) or just used to state a rule.

Il te faut faire la vaisselle is a rare way to say the same. It sounds literary, and is more expected in a book than in a casual conversation. It might also be heard as a regionalism in Southern France, a reminiscence of the occitan te cal faire la vaissèla

Here is an example of this regionalism, posted on a forum by someone from the Hérault (Occitanie):

comme disait lefab, il te faut choisir qu’elle couleur tu préfere et aprés seulement on pourra t’aider parce que la je peut t’en proposé des centaines. ^{_{(original spelling and grammar untouched)}}

Il faut faire la vaisselle is a generic way to say doing the dishes is required.