The IBOR forecast curve is the market's expectation of the evolution of IBOR fixings. If today is 1-Jan-2018 and you have a curve that prices 27-April-2018 for 3M as 1% then if the market evolves as predicted and there is no (exogenous supply/demand) market movement between today and tomorrow then on the 2-Jan-2018 the 3M rate priced to start at 27-April-2018 is the same as it was yesterday, i.e. 1%. This is precisely what you will observe on futures markets, since the futures contract settle to fixed dates (IMM dates). Any fluctuation in those rates is market movement, i.e. change in expectations.
However if your dates are generic maturities, i.e. start-3M-end-3M, then the rates will change since a forward rate that is priced as 3M3M from 1-Jan-2018 has dates 1-April-2018 to 1-Jul-28, whereas a 3M3M priced from 2-Jan-2018 has dates 2-April-18 to 2-Jul-18, and therefore it does not represent the same price as yesterday.