You can just round the result, and then they are the same number.  
    
    Total[E^(Round[list, 10.^-6]/t)]
    (* 4 E^(-0.683013/t) + 3 E^(-0.183013/t) + 
     2 E^(0.183013/t) + E^(0.683013/t) *)

The `Exp` and `Round` function are listable, so you don't need an explicit call to `Sum`.  But if you wanted one, 

    Sum[ E^(Round[list[[k]], 10.^-6]/t), {k, Length[list]}]

**Edit** You can also use `SetPrecision`
    
    Total@Exp[ SetPrecision[list, 6]/t] 

    (* 4 E^(-0.683013/t) + 3 E^(-0.183013/t) + 
     2 E^(0.183013/t) + E^(0.683013/t) *)

But I don't see an equivalent to `SameTest` for `Plus`