data = {{0, 5678}, {0, 5654}, {1, 87675},
{2, 243}, {2, 257}, {1, 9790},
{0, 7688}, {1, 97779}, {0, 6858}};
Using SequencePosition and Extract:
f = Extract[#1, List /@ Union @@ SequencePosition[#1, {{#2, _}, {#3, _}}]] &;
f[data, 0, 1]
{{0, 5654}, {1, 87675}, {0, 7688}, {1, 97779}}
f[data, 1, 2]
{{1, 87675}, {2, 243}}
Or using Partition and Pick:
f[l_, {m_, n_}] :=
Join @@ Pick[#, #[[1, 1]] == m && #[[2, 1]] == n & /@ #] &@Partition[l, 2, 1]
f[data, {0, 1}]
{{0, 5654}, {1, 87675}, {0, 7688}, {1, 97779}}
f[data, {1, 2}]
{{1, 87675}, {2, 243}}