When you're talking about a general state of affairs, you should use "tá".
e.g. "Tá an teach mór" doesn't mean that the house is big just at this moment in time.
When you're talking about specific occasions at separate points in time (the sort of situations in which you might use words like "go minic", "de ghnáth", "gach lá", "uaireanta"), you use "bíonn".
e.g. "Tá sé cúramach" can imply that he is a careful person in general or that he is being careful at this precise moment, depending on context; "Bíonn sé cúramach", on the other hand, means that he is regularly careful at various different moments: "Bíonn sé cúramach agus é i mbun siosúir" = He is careful when (each time) he uses scissors.
So, there is no class of adjectives which use "bíonn" rather than "tá", more or less any adjective can take either tense depending on the timeframe you want to express. You'll generally need to use "tá", unless you're specifically talking about a number of different occasions.
As well as "tá" and "bíonn", the copula ("is") is also regularly used with adjectives, generally when the adjective is emphasised: "Is mór an teach sin", "Is cúramach an duine é".