Can the list price of a house be changed after an offer is made?

Unlike when you go into say, a department store and see something listed as for sale for $31.00, and the store *must* take your $31.00 and sell you that item, a home-seller has the right to turn down any offer--even a market price or above market price offer--for any reason that is not a prohibited form of discrimination (the main one of which, for housing purposes, is racial discrimination). They can take a lower offer from person A while insisting on more money from person B. So as a general matter, this is legal.
It is further legal because you wanted closing fees: that would cost them money and effectively reduces the price of the house. For example, say that closing costs may be $5k - $6k. An offer of $310k but seller pays closing is really the equivalent of a $304k to $305k offer, because paying closng reduces what they get by $5-6k. 
Evidently, they want to get $310k for their house--not $310k less closing. So they came back and said sure--pay us $316k and we'll pay your closing, so we'll net out to around $310k after closing. That is both legal and reasonable: they don't have to accept less profit on their house because you want your closing costs paid.
When you wouldn't do that, they went back to listing the house at the same $310k it was at and they wanted--that is legal.
FYI, I don't know your real estate market, but if it's anything like mine (northern NJ), try making a non-closing cost offer: in my area, where houses sell well, no sellers accept an offer requiring them to pay closing costs unless the offer is well over asking price.