[Timeless Thinker]

25 mL of 0.10 M acetic acid is titrated with 0.10 M NaOH. What is the pH after 5 mL of NaOH have been added? Ka for acetic acid = 1.8 × 10–5.
So I created an ICE table in moles of each substance:
CH₃COOH +    OH^-           CH₃COO^-
I:  .0025        .0005                   
C: -.0005      -.0005           +.0005
E: .0020         0                    .0005

Then I converted these back to M using n/V
E: .0667         0                   .0167

After this, I used Henderson Hasselbalch equation of pH= pKa+ log[A^-]/[HA]
4.745+ -.602= 4.142
pH= 4.142
 
Should I be getting a negative number for my log[A^-]/[HA]?

[Borek]

25 mL of 0.10 M acetic acid is titrated with 0.10 M NaOH. What is the pH after 5 mL of NaOH have been added? Ka for acetic acid = 1.8 × 10–5.
So I created an ICE table in moles of each substance:
CH₃COOH +    OH^-           CH₃COO^-
I:  .0025        .0005                   
C: -.0005      -.0005           +.0005
E: .0020         0                    .0005

That's not entirely correct. The problem here is how one defines Initial, Change and Equilibrium. You got a correct stoichiometry - and that's a right first step - but not E (equilibrium) concentrations. Equilibrium after the neutralization can slightly shift (and you could use ICE table to find that shift). Usually the change is negligible, compare https://www.chembuddy.com/?left=buffers&right=with-ICE-table

Then I converted these back to M using n/V

Correct, but not necessary - you divided nominator and denominator by the same number, it cancels out.

After this, I used Henderson Hasselbalch equation of pH= pKa+ log[A^-]/[HA]
4.745+ -.602= 4.142
pH= 4.142

Looks OK.
 

Should I be getting a negative number for my log[A^-]/[HA]?

Why not? Sometimes it will be negative, sometimes positive.

[Timeless Thinker]

But it's being titrated with a strong base, why would the pH go down?

[AWK]

Goes down?
pH of 0.1 M acetic acid is close to 2.9.

[Timeless Thinker]

But when I take the log of [A-]/[HA], is -.602, is that correct?

[AWK]

You diminished pKa value, not pH.

[Borek]

But when I take the log of [A-]/[HA], is -.602, is that correct?

Yes.

Your starting point is not [HA]=[A^-], in a 0.1M solution of acetic acid initial [A^-]/[HA] ratio is around 0.01 - and the log is close to -2.