There are a couple of errors along the way.
test1 = {{"A", 110468., 1978., 0.142501846}, {"A", 110512., 1978.,
0.171648613}, {"A", 110540., 1978., 0.121514693}, {"A", 111721.,
1978., 0.218872092}, {"A", 111722., 1978., 0.083322812}, {"A",
111725., 1978., 0.120864387}, {"A", 111728., 1978.,
0.139541269}, {"A", 111916., 1978., 0.285463369}, {"A", 110468.,
1979., 0.542969073}, {"A", 110512., 1979., 0.192196573}};
First is that Cases is pattern-match based, in contrast to the functional criterion Select. So you can do like so.
DeleteCases[test1, aa_ /; aa[[2]] == 110468]
(* Out[5]= {{"A", 110512., 1978., 0.171649}, {"A", 110540., 1978.,
0.121515}, {"A", 111721., 1978., 0.218872}, {"A", 111722., 1978.,
0.0833228}, {"A", 111725., 1978., 0.120864}, {"A", 111728., 1978.,
0.139541}, {"A", 111916., 1978., 0.285463}, {"A", 110512., 1979.,
0.192197}} *)
For the multiple criteria use of Select, you want And rather than Or to get the correct logic.
Select[test1, #[[2]] != 110468 && #[[2]] != 110512 &]
(* Out[8]= {{"A", 110540., 1978., 0.121515}, {"A", 111721., 1978.,
0.218872}, {"A", 111722., 1978., 0.0833228}, {"A", 111725., 1978.,
0.120864}, {"A", 111728., 1978., 0.139541}, {"A", 111916., 1978.,
0.285463}} *)
Also there is the Wolfram Function Repository resource function Discard that can be used.
ResourceFunction["Discard"][test1, #[[2]] == 110468 &]
(* Out[11]= {{"A", 110512., 1978., 0.171649}, {"A", 110540., 1978.,
0.121515}, {"A", 111721., 1978., 0.218872}, {"A", 111722., 1978.,
0.0833228}, {"A", 111725., 1978., 0.120864}, {"A", 111728., 1978.,
0.139541}, {"A", 111916., 1978., 0.285463}, {"A", 110512., 1979.,
0.192197}} *)
