I have found numerous examples where people reference Psalms to justify their theological positions. However, because of the unique genre and purpose of Psalms, is it acceptable to do so? If so, how do we then reconcile Psalms like psalm 137:9?
Psalms is quoted many times in the NT. As seen by an earlier post Jesus quoted Psalms more than any other book.
2:1, 2 Acts 4:25, 26
2:7 Acts 13:33; Hebrews 1:5 ; 5:5
2:8, 9 Revelation 2:26, 27; 12:5; 19:15
4:4 Ephesians 4:26
5:9 Romans 3:13
6:3a John 12:27
6:8 Matthew 7:23; Luke 13:27
8:2 Matthew 21:16
8:4-6 Hebrews 2:6-8
8:6 1 Corinthians 15:27; Ephesians 1:22
10:7 Romans 3:14
14:1c, 2b, 3 Romans 3:10-12
16:8.-11 Acts 2:25-28
16:10b Acts 13:35
18:2b Hebrews 2:13
18:49 Romans 15:9
19:4 Romans 10:18
22:1 Matthew 27:46; Mark 15:34
22:7 Matthew 27:39; Mark 15:29; Luke 23:35
22:8 Matthew 27:43
22:18 John 19:24; compare Matthew 27:35; Mark 15:24; Luke 23:34
22:22 Hebrews 2:12
24:1 1 Corinthians 10:26 [28]
31:5a Luke 23:46
32:1, 2 Romans 4:7, 8
34:8 1 Peter 2:3
34:12-16 1 Peter 3:10-12
34:20 John 19:36
35:19b John 15:25
36:1b Romans 3:18
37:11a Matthew 5:5
38:11 Luke 23:49
40:6-8 Hebrews 10:5-7
41:9 John 13:18
41:13 Luke 1:68
42:5 Matthew 26:38; Mark 14:34
44:22 Romans 8:36
45:6, 7 Hebrews 1:8, 9
48:2 Matthew 5:35
51:4 Romans 3:4
53:1-3 Romans 3:10-12
55:22 1 Peter 5:7
62:12 Matthew 16:27; Romans 2:6
68:18 Ephesians 4:8
69:4 John 15:25
69:9a John 2:17
69:9b Romans 15:3
69:21 Matthew 27:34, 48; Mark 15:36; Luke 23:36; John 19:28, 29
69:22, 23 Romans 11:9, 10
69:25 Acts 1:20
72:18 Luke 1:68
78:2 Matthew 13:35
78:24 John 6:31
82:6 John 10:34
86:9 Revelation 15:4
88:8 Luke 23:49
89:10 Luke 1:51
89:20 Acts 13:22
90:4 2 Peter 3:8
91:11, 12 Matthew 4:6; Luke 4:10, 11
91:13 Luke 10:19
94:11 1 Corinthians 3:20
94:14 Romans 11:1, 2
95:7-11 Hebrews 3:7-11, 15, 18; 4:1, 3, 5, 7
97:7 Hebrews 1:6
98:3 Luke 1:54
102:25-27 Hebrews 1:10-12
103:17 Luke 1:50
104:4 Hebrews 1:7
105:8, 9 Luke 1:72, 73
106:10 Luke 1:71
106:45 Luke 1:72
106:48 Luke 1:68
107:9 Luke 1:53
109:8 Acts 1:20
109:25 Matthew 27:39
110:1 Matthew 22:44; Mark 12:36; Luke 20:42, 43; Acts 2:34, 35; Hebrews 1:13. Compare. Matthew 26:64; Mark 14:62; 16:19; Luke 22:69; 1 Corinthians 15:25; Ephesians 1:20; Colossians 3:1; Hebrews 1:3; 8:1; 10:12, 13; 12:2; 1 Peter 3:22
110:4 Hebrews 5:6; 6:20; 7:17, 21
111:9a Luke 1:68
111:9c Luke 1:49
112:9 2 Corinthians 9:9
116:10 2 Corinthians 4:13
117:1 Romans 15:11
118:6 Hebrews 13:6
118:22, 23 Matthew 21:42; Mark 12:10, 11; Luke 20:17; Acts 4:11; 1 Peter 2:4, 7
118:25, 26 Matthew 21:9; 23:39; Mark 11:9; Luke 13:35; 19:38; John 12:13
132:5 Acts 7:46
132:11 Acts 2:30
132:17 Luke 1:69
135:14a Hebrews 10:30
140:3b Romans 3:13
143:2b Romans 3:20
146:6 Acts 4:24; 14:15