NPN Transistor voltage calculation
A) Vbe is used to calculate Ib  and thus Vce = V+ - βIbRc is in the linear region

B) As  β  goes to  ∞ and Ib goes to 0  then it no longer is linear and Vce is saturated where in this case Vce(sat) = Vbe  or Vbc=0  so different rules apply.

Side note

B) is purely hypothetical , because when Vce goes towards saturation β drops as much as 10% of the linear value and datasheets rate Vce(sat) for some rated current.   Vce(sat)=0.1V is never reached with the negative feedback from Collector to base or ANY value of β.