I tried my hand at some data collection. I think it's likely that the game does not run faster when line mode is activated.
-----
The theory: Run streets agent a bunch of times with and without line mode. Given enough data, a statistically relevant result will emerge, assuming the runs are consistent.
Data collection: I ran streets 100 times on agent (details below). I used a gameshark to expose the line mode cheat and completed another 100 runs. (I streamed some of this and met a few elite members). For various reasons, I wasn't really happy with this data set, but the preliminary data showed that line mode was a bit slower. I changed some of my setup and collected a second data set, 50 runs normal and 50 runs with line mode. I was much happier with the quality of this data set (variance is much lower), but once again this seemed to indicate that line mode is much slower. For reference, the 50 run result difference of averages is 0.88 seconds.
Technical Discussion: I think the assumption in this thread is that less stuff is being drawn on screen, and therefore there is less processing going on, and therefore there will be less lag. I had also assumed enabling line mode would make the game slightly faster, but since the data collection seems to disagree, I think it's important to find a technical explanation (or an explanation for why the data is bad). If you check the Goldeneye n64decomp project, you can find where the linemode check happens in the main game loop: https://github.com/n64decomp/007/blob/045df043162ff5dac7c324e9cf74f3d877067187/src/boss.c#L504 (https://github.com/n64decomp/007/blob/045df043162ff5dac7c324e9cf74f3d877067187/src/boss.c#L504). (You can test this by replacing the branch instruction at 0x70006620 (/* 007220 70006620 10400037 */ beqz $v0, .L70006700) with a nop). Interestingly, the Project64 default graphics plugin does not handle this gracefully, but it does kind of visualize what's happening, in that basically everything on screen gets drawn again, but this time just showing the edges (the GlideN64 plugin works correctly for linemode). To summarize: when line mode is enabled, there's extra work the main game loop does which causes additional graphics rendering.
Data Collection Limitations: the elite members that stopped by my stream were concerned about my game skill level and offered some helpful advice (this is partly why I threw out the first data set). I say that as a preface, that when line mode is active it's harder to orient yourself in the level. If you're strafing against the wall, you can't watch a texture scroll by to gauge your position because the screen is just white. So, it's possible that someone more skilled at the game would see line mode times closer to normal time. On the other hand, line mode feels laggier, but I don't know how much of that effect is because the black/white divisions on screen are "sharper" than just normal textures.
Data Collection Limitations (boosts): I did try to track boost counts with this data set, but .... What counts as a boost? (some rhetorical questions now) Getting shot parallel to your current trajectory? What about getting shot perpendicular, so your route is longer? What if you're boosted forward, but your route is longer? I tried to lump anything that improved your time as a forward boost, and anything that made your run take longer as a back boost. For especially bad back boosts (e.g., getting shot into a corner) I would reset the run. Overall, the normal run averaged 2.18 forward and 2.22 back. Line mode averaged 1.88 forward and 2.38 back. Does 0.46 boost difference account for 0.88 seconds? I don't think so, but ...
Technical Discussion Limitations: I'm not a N64 graphics expert, my above explanation may not be correct.
Conclusion: Line mode is probably slightly slower than normal graphics mode.
-----
Appendix A - setup
Spoiler
controller 2.3
NTSC
US
no lookdown (always the neutral position)
WR (street) route, full BAx2 route, no self boost/grenades/rockets
music max
fx max
auto-aim on
sight on-screen on
look ahead on
ammo on-screen on
screen full
ratio normal
CC move 43
CC strafe 73
Appendix B - regular run data
Spoiler
#,game time,boost forward,boost back
1,1:20,1,4
2,1:19,3,3
3,1:19,5,1
4,1:19,3,1
5,1:18,2,2
6,1:19,3,0
7,1:19,3,2
8,1:21,1,2
9,1:19,2,3
10,1:18,2,1
11,1:19,2,2
12,1:20,1,3
13,1:18,0,1
14,1:19,2,3
15,1:20,1,3
16,1:19,2,0
17,1:19,1,3
18,1:21,2,6
19,1:20,4,4
20,1:19,2,1
21,1:21,3,7
22,1:20,2,2
23,1:19,3,2
24,1:19,2,2
25,1:20,1,5
26,1:20,1,2
27,1:20,4,6
28,1:19,1,2
29,1:19,1,3
30,1:20,2,5
31,1:19,1,2
32,1:19,2,2
33,1:19,2,1
34,1:18,2,0
35,1:18,2,1
36,1:17,4,1
37,1:19,3,1
38,1:19,2,0
39,1:19,3,1
40,1:18,4,3
41,1:19,1,1
42,1:19,3,2
43,1:18,1,1
44,1:18,3,0
45,1:18,1,0
46,1:19,3,4
47,1:19,2,0
48,1:19,3,5
49,1:18,4,4
50,1:19,1,1
Appendix C - line mode run data
Spoiler
#,game time,boost forward,boost back
1,1:20,0,2
2,1:20,2,3
3,1:19,1,2
4,1:20,4,4
5,1:20,3,4
6,1:20,1,2
7,1:19,4,1
8,1:20,0,2
9,1:20,4,2
10,1:19,2,0
11,1:20,2,2
12,1:21,3,4
13,1:19,0,0
14,1:20,4,2
15,1:21,1,2
16,1:19,3,2
17,1:22,1,3
18,1:20,1,0
19,1:20,3,2
20,1:21,5,3
21,1:20,1,1
22,1:20,1,0
23,1:20,3,0
24,1:19,3,3
25,1:18,1,2
26,1:20,0,2
27,1:21,0,1
28,1:21,2,4
29,1:20,0,3
30,1:19,3,4
31,1:19,4,4
32,1:20,2,2
33,1:20,2,4
34,1:20,2,5
35,1:19,1,1
36,1:20,0,1
37,1:20,3,3
38,1:20,0,2
39,1:19,3,6
40,1:22,2,2
41,1:21,2,5
42,1:21,1,4
43,1:19,1,2
44,1:19,1,1
45,1:20,3,4
46,1:20,3,3
47,1:20,2,4
48,1:19,1,0
49,1:21,2,4
50,1:20,1,0