Find longest sequence of zeros in list

Assuming it is a binary sequence, one way is to look at all the differences in the positions of the 1s and take the largest:

seq = {1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1};
Max[Differences[Position[seq, 1]]] - 1
4

As pointed out in the comments by jkuczm and Hector, this will fail if the longest sequence of zeros occurs at either the start or the end. This can be fixed by surrounding the sequence with 1s:

Max[Differences[Position[Flatten@{1, seq, 1}, 1]]] - 1

Max@FoldList[If[#2 == 0, #1 + 1, 0] &, 0, seq]

If you need speed and are only using packable arrays you could use a compiled function. Function generator returning compiled functions, performing desired operation can look like this:

compileLongestSeqLen // ClearAll
compileLongestSeqLen[
  {type_, rank_Integer /; rank >= 1}, inSeqQ_, opts : OptionsPattern@Compile
] := Compile[{{list, type, rank}},
  Module[{len = 0, longest = 0},
    Do[
      If[inSeqQ@x,
        ++len;
      (* else *),
        If[len > longest, longest = len];
        len = 0;
      ]
      ,
      {x, list}
    ];
    If[len > longest, longest = len];
    longest
  ],
  opts
]

Let's compile function finding length of longest sequence of zeros in list of integers:

jkuczm = compileLongestSeqLen[{_Integer, 1}, # === 0 &,
  RuntimeOptions -> "Speed", CompilationTarget -> "C",
  RuntimeAttributes -> {Listable}, Parallelization -> True
]

To compare efficiency let's define functions from other answers:

alan = Max@FoldList[If[#2 == 0, #1 + 1, 0] &, 0, #] &;
billSHect1 = Max[Differences[Position[Flatten@{1, #, 1}, 1]]] - 1 &;
billSHect2 = Max[Differences[Position[Join[{1}, #, {1}], 1]]] - 1 &;
carlL = # /. {___, a : Longest[Repeated[0]], ___} :> Length[{a}] &;
chyanog1 = GroupBy[Split[#], First -> Length, Max][0] &;
chyanog2 = (# // Split // Select[MemberQ[0]] // Map[Length] // Max) &;
joe = (SortBy[Split[#], Length] // Last // Length) &;
kglr1 = Max[Differences@PositionIndex[Join[{1}, #, {1}]][1]] - 1 &;
kglr2 = Max[Length /@ Split@Accumulate[Join[{1}, #, {1}]]] - 1 &;
maxZeros[a_List] := Max[Append[# - 1, Length@a] - Prepend[#, 0]] &@SparseArray[a]["AdjacencyLists"]
okkes = (Length /@ Split[#] // Max) &;

Timings for single large list:

SeedRandom@0
test = RandomInteger[{0, 1}, 10^6];

(*carlLRes = carlL@test; // MaxMemoryUsed // RepeatedTiming*)   (*too slow*)
chyanog2Res = chyanog2@test; // MaxMemoryUsed // RepeatedTiming (*{0.62,  44510360}*)
(*joeRes = joe@test; // MaxMemoryUsed // RepeatedTiming*)       (*incorrect results*)
chyanog1Res = chyanog1@test; // MaxMemoryUsed // RepeatedTiming (*{0.36,  64019000}*)
billSRes1 = billSHect1@test; // MaxMemoryUsed // RepeatedTiming (*{0.33, 119935248}*)
billSRes2 = billSHect2@test; // MaxMemoryUsed // RepeatedTiming (*{0.31, 119934952}*)
kglr2Res = kglr2@test; // MaxMemoryUsed // RepeatedTiming       (*{0.26,  75941464}*)
(*okkesRes = okkes@test; // MaxMemoryUsed // RepeatedTiming*)   (*incorrect results*)
kglr1Res = kglr1@test; // MaxMemoryUsed // RepeatedTiming       (*{0.11,  44568024}*)
alanRes = alan@test; // MaxMemoryUsed // RepeatedTiming         (*{0.063, 40008744}*)
mrWizRes = maxZeros@test; // MaxMemoryUsed // RepeatedTiming    (*{0.0273 17126232}*)
jkuczmRes = jkuczm@test; // MaxMemoryUsed // RepeatedTiming     (*{0.0047,      56}*)

(*carlLRes ===*) chyanog2Res === (*joeRes ===*) chyanog1Res === billSRes1 === billSRes2 === kglr2Res === (*okkesRes ===*) kglr1Res === alanRes === mrWizRes === jkuczmRes
(* True *)

Timings for multiple lists:

SeedRandom@0
test = RandomInteger[{0, 1}, {100, 10^5}];

(*carlLRes=carlL/@test;//MaxMemoryUsed//RepeatedTiming*)           (*too slow*)
chyanog2Res = chyanog2 /@ test; // MaxMemoryUsed // RepeatedTiming (*{6.3,  84491416}*)
(*joeRes = joe /@ test; // MaxMemoryUsed // RepeatedTiming*)       (*incorrect results*)
chyanog1Res = chyanog1 /@ test; // MaxMemoryUsed // RepeatedTiming (*{3.46, 86460488}*)
billSRes1 = billSHect1 /@ test; // MaxMemoryUsed // RepeatedTiming (*{3.29, 92106568}*)
kglr2Res = kglr2 /@ test; // MaxMemoryUsed // RepeatedTiming       (*{2.68, 87630888*)
(*okkesRes = okkes /@ test; // MaxMemoryUsed // RepeatedTiming*)   (*incorrect results*)
kglr1Res = kglr1 /@ test; // MaxMemoryUsed // RepeatedTiming       (*{1.12, 84780440}*)
alanRes = alan /@ test; // MaxMemoryUsed // RepeatedTiming         (*{0.379, 2409680}*)
billSRes2 = billSHect2 /@ test; // MaxMemoryUsed // RepeatedTiming (*{0.24,  2400320}*)
mrWizRes = maxZeros /@ test; // MaxMemoryUsed // RepeatedTiming    (*{0.22, 81755512}*)
jkuczmRes = jkuczm@test; // MaxMemoryUsed // RepeatedTiming        (*{0.030, 1602456}*)

(*carlLRes ===*) chyanog2Res === (*joeRes ===*) chyanog1Res === billSRes1 === kglr2Res === (*okkesRes ===*) kglr1Res === alanRes === billSRes2 === mrWizRes === jkuczmRes
(* True *)