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CJam, 33 bytes

La'|ali({_'|f+@"——"f++}*\;{N+_N}/

This is an iterative implementation of a recursive algorithm:

The possible tilings for n can be obtained by adding a vertical domino to the possible tilings for n - 1 and a horizontal domino to the possible tilings for n - 2. This way, the number of tilings for n is the sum of the numbers of tilings for n - 1 and n - 2, i.e., the n^th Fibonacci number.

Try it online.

How it works

La'|a " A:= [''] B:= ['|'] ";
 li({ }* " Repeat int(input()) - 1 times: ";
 _'|f+ " C = copy(B); for T ∊ C: C += '|' ";
 @ " Swap A and B. ";
 "--"f+ " for T ∊ B: T += '--' ";
 + " B = C + B ";
 \; " Discard A. ";
 {N+_N}/ " for T ∊ B: print T + '\n', T + '\n', '\n' ";

Example run

$ cjam domino.cjam <<< 3
|||
|||
——|
——|
|——
|——
$ for i in {1..10}; do echo $[$(cjam domino.cjam <<< $i | wc -l) / 3]; done
1
2
3
5
8
13
21
34
55
89