I examined the 2016 results. There 155,000 precincts with at least 200 votes.
Eighty-nine such precincts had <3% Democratic votes. The majority were in the Deep South.
State | Precincts
-------+-----------
AL | 39
AZ | 1
GA | 3
ID | 1
KS | 1
KY | 1
LA | 15
MS | 10
MT | 1
NE | 1
NJ | 2
SC | 3
TX | 2
UT | 5
WY | 4
The twenty least Democratic precincts were:
State | County | Precinct | D | R | Total
-------+---------------------+----------------------+----+-----+-------
AL | Washington County | COPELAND ASSEMBLY OF | 1% | 78% | 382
MS | Neshoba County | Fusky | 1% | 98% | 331
AL | Crenshaw County | PATSBURG DEPOT BLDG_ | 1% | 73% | 216
NJ | Ocean County | Lakewood District 36 | 1% | 93% | 292
AL | Choctaw County | LUSK | 1% | 73% | 266
MT | Ravalli County | PCT 14 | 1% | 90% | 237
MS | Greene County | Piave | 1% | 96% | 314
WY | Campbell County | Wright Town Hall 14- | 1% | 97% | 350
AL | Choctaw County | BARRYTOWN | 1% | 72% | 280
LA | LaSalle Parish | 09 01 | 1% | 98% | 276
AL | Fayette County | LEE-OAKRIDGE VOTING | 1% | 64% | 205
MS | Lamar County | Baxterville | 2% | 97% | 348
SC | Williamsburg County | PINEY FOREST | 2% | 57% | 402
AL | Lamar County | VAILS | 2% | 60% | 341
AL | Winston County | MORELAND FIRE DEPT | 2% | 58% | 279
WY | Crook County | Moorcroft Outside - | 2% | 94% | 332
LA | Vernon Parish | 06 04 | 2% | 96% | 276
GA | Appling County | 3A | 2% | 96% | 217
TX | Panola County | 3650020 | 2% | 95% | 215
TX | Carson County | 650402 | 2% | 97% | 211
All precincts of that minimum size had at least 1 Democratic vote.
For comparison, 4802 precincts had <3% Republican votes.
Twenty-five had exactly zero Republican votes, including:
State | County | Precinct | D | R | Total
-------+----------------------+----------------------+------+----+-------
AL | Mobile County | THOMAS SULLIVAN COMM | 53% | 0% | 532
IL | Cook County | Ward 20 Precinct 12 | 99% | 0% | 438
IL | Cook County | Ward 24 Precinct 35 | 99% | 0% | 438
OH | Cuyahoga County | CLEVELAND-02-R | 99% | 0% | 425
IL | Cook County | Ward 09 Precinct 05 | 99% | 0% | 423
PA | Philadelphia County | 2170 | 100% | 0% | 406
LA | Pointe Coupee Parish | 00 13A | 99% | 0% | 392
PA | Philadelphia County | 2830 | 100% | 0% | 350
PA | Philadelphia County | 5890 | 99% | 0% | 312
AL | Wilcox County | PORTABLE BUILDING | 52% | 0% | 303
LA | St. James Parish | 00 07 | 100% | 0% | 292
PA | Philadelphia County | 1230 | 97% | 0% | 285
MD | Baltimore city | School No. 35|pplace | 98% | 0% | 283
PA | Philadelphia County | 7090 | 100% | 0% | 282
PA | Philadelphia County | 6230 | 99% | 0% | 280
PA | Philadelphia County | 3430 | 99% | 0% | 268
NY | Kings County | 056/56 | 94% | 0% | 254
IN | Lake County | G4-13 | 99% | 0% | 231
AL | Wilcox County | LOWER PEACHTREE DIST | 54% | 0% | 220
AR | Desha County | MITCHELLVILLE | 59% | 0% | 219
(Interestingly, Louisiana has two of the 20 least Democratic precincts and two of the 20 least Republican precincts.)
It's unclear how you want to treat independents,who played an unusually large role in 2016. FYI, only two precincts with >=200 votes actually had only Democratic votes.
There is a well-known phenomenon that Democratic votes are significantly more clustered. This gives a slight "natural gerrymandering" advantage to Republicans.