diff --git a/ci/scripts/r_test.sh b/ci/scripts/r_test.sh
index d447bdf23..1f6a8721e 100755
--- a/ci/scripts/r_test.sh
+++ b/ci/scripts/r_test.sh
@@ -37,10 +37,17 @@ if [ "$ARROW_R_DEV" = "TRUE" ]; then
   # Note that NOT_CRAN=true means (among other things) that optional dependencies are built
   export NOT_CRAN=true
 fi
-export TEST_R_WITH_ARROW=TRUE
-export _R_CHECK_TESTS_NLINES_=0
+: ${TEST_R_WITH_ARROW:=TRUE}
+export TEST_R_WITH_ARROW=$TEST_R_WITH_ARROW
+
 export _R_CHECK_CRAN_INCOMING_REMOTE_=FALSE
+# --run-donttest was used in R < 4.0, this is used now
+export _R_CHECK_DONTTEST_EXAMPLES_=$TEST_R_WITH_ARROW
+# Not all Suggested packages are needed for checking, so in case they aren't installed don't fail
+export _R_CHECK_FORCE_SUGGESTS_=FALSE
 export _R_CHECK_LIMIT_CORES_=FALSE
+export _R_CHECK_TESTS_NLINES_=0
+
 # By default, aws-sdk tries to contact a non-existing local ip host
 # to retrieve metadata. Disable this so that S3FileSystem tests run faster.
 export AWS_EC2_METADATA_DISABLED=TRUE
@@ -49,9 +56,6 @@ export AWS_EC2_METADATA_DISABLED=TRUE
 export TEXMFCONFIG=/tmp/texmf-config
 export TEXMFVAR=/tmp/texmf-var
 
-# Not all Suggested packages are needed for checking, so in case they aren't installed don't fail
-export _R_CHECK_FORCE_SUGGESTS_=FALSE
-
 if [[ "$DEVTOOLSET_VERSION" -gt 0 ]]; then
   # enable the devtoolset version to use it
   source /opt/rh/devtoolset-$DEVTOOLSET_VERSION/enable
@@ -61,8 +65,9 @@ fi
 BEFORE=$(ls -alh ~/)
 
 SCRIPT="as_cran <- !identical(tolower(Sys.getenv('NOT_CRAN')), 'true')
+  run_donttest <- identical(tolower(Sys.getenv('_R_CHECK_DONTTEST_EXAMPLES_', 'true')), 'true')
   if (as_cran) {
-    rcmdcheck::rcmdcheck(args = c('--as-cran', '--run-donttest'), error_on = 'warning', check_dir = 'check', timeout = 3600)
+    rcmdcheck::rcmdcheck(args = c('--as-cran', if (run_donttest) '--run-donttest'), error_on = 'warning', check_dir = 'check', timeout = 3600)
   } else {
     if (nzchar(Sys.which('minio'))) {
       message('Running minio for S3 tests (if build supports them)')
@@ -71,7 +76,7 @@ SCRIPT="as_cran <- !identical(tolower(Sys.getenv('NOT_CRAN')), 'true')
       pid <- sys::exec_background('minio', c('server', minio_dir))
       on.exit(tools::pskill(pid))
     }
-    rcmdcheck::rcmdcheck(build_args = '--no-build-vignettes', args = c('--no-manual', '--ignore-vignettes', '--run-donttest'), error_on = 'warning', check_dir = 'check', timeout = 3600)
+    rcmdcheck::rcmdcheck(build_args = '--no-build-vignettes', args = c('--no-manual', '--ignore-vignettes', if (run_donttest) '--run-donttest'), error_on = 'warning', check_dir = 'check', timeout = 3600)
   }"
 echo "$SCRIPT" | ${R_BIN} --no-save
 
diff --git a/dev/tasks/r/azure.linux.yml b/dev/tasks/r/azure.linux.yml
index 4c977fe20..6e022ec04 100644
--- a/dev/tasks/r/azure.linux.yml
+++ b/dev/tasks/r/azure.linux.yml
@@ -47,8 +47,7 @@ jobs:
           export R_ORG={{ r_org }}
           export R_IMAGE={{ r_image }}
           export R_TAG={{ r_tag }}
-          # we have to export this (right?) because we need it in the build env
-          export ARROW_R_DEV={{ not_cran }}
+          export ARROW_R_DEV={{ not_cran|default("TRUE") }}
           # Note that ci/scripts/r_test.sh sets NOT_CRAN=true if ARROW_R_DEV=TRUE
           docker-compose run \
             -e ARROW_DATASET={{ arrow_dataset|default("") }} \
@@ -57,6 +56,9 @@ jobs:
             -e ARROW_WITH_RE2={{ arrow_with_re2|default("") }} \
             -e ARROW_WITH_UTF8PROC={{ arrow_with_utf8proc|default("") }} \
             -e LIBARROW_MINIMAL={{ libarrow_minimal|default("") }} \
+            -e LIBARROW_DOWNLOAD={{ libarrow_download|default("") }} \
+            -e LIBARROW_BUILD={{ libarrow_build|default("") }} \
+            -e TEST_R_WITH_ARROW={{ with_arrow|default("TRUE") }} \
             r
         displayName: Docker run
 
diff --git a/dev/tasks/tasks.yml b/dev/tasks/tasks.yml
index a3f05a82a..84114c0d2 100644
--- a/dev/tasks/tasks.yml
+++ b/dev/tasks/tasks.yml
@@ -1763,7 +1763,6 @@ tasks:
       r_org: rhub
       r_image: ubuntu-gcc-release
       r_tag: latest
-      not_cran: "TRUE"
 
   test-r-rocker-r-base-latest:
     ci: azure
@@ -1772,7 +1771,6 @@ tasks:
       r_org: rocker
       r_image: r-base
       r_tag: latest
-      not_cran: "TRUE"
 
   test-r-rstudio-r-base-3.6-bionic:
     ci: azure
@@ -1781,7 +1779,6 @@ tasks:
       r_org: rstudio
       r_image: r-base
       r_tag: 3.6-bionic
-      not_cran: "TRUE"
 
   test-r-rstudio-r-base-3.6-centos8:
     ci: azure
@@ -1790,7 +1787,6 @@ tasks:
       r_org: rstudio
       r_image: r-base
       r_tag: 3.6-centos8
-      not_cran: "TRUE"
 
   test-r-rstudio-r-base-3.6-centos7-devtoolset-8:
     ci: azure
@@ -1799,7 +1795,6 @@ tasks:
       r_org: rstudio
       r_image: r-base
       r_tag: 3.6-centos7
-      not_cran: "TRUE"
       devtoolset_version: 8
 
   test-r-rstudio-r-base-3.6-opensuse15:
@@ -1809,7 +1804,6 @@ tasks:
       r_org: rstudio
       r_image: r-base
       r_tag: 3.6-opensuse15
-      not_cran: "TRUE"
 
   test-r-rstudio-r-base-3.6-opensuse42:
     ci: azure
@@ -1818,7 +1812,6 @@ tasks:
       r_org: rstudio
       r_image: r-base
       r_tag: 3.6-opensuse42
-      not_cran: "TRUE"
 
   test-r-minimal-build:
     ci: azure
@@ -1827,7 +1820,6 @@ tasks:
       r_org: rocker
       r_image: r-base
       r_tag: latest
-      not_cran: "TRUE"
       arrow_dataset: "OFF"
       arrow_parquet: "OFF"
       arrow_s3: "OFF"
@@ -1835,6 +1827,18 @@ tasks:
       arrow_with_utf8proc: "OFF"
       libarrow_minimal: "TRUE"
 
+  test-r-without-arrow:
+    ci: azure
+    template: r/azure.linux.yml
+    params:
+      r_org: rhub
+      r_image: ubuntu-gcc-release
+      r_tag: latest
+      libarrow_download: "FALSE"
+      libarrow_build: "FALSE"
+      with_arrow: "FALSE"
+      not_cran: "FALSE"
+
   test-ubuntu-18.04-r-sanitizer:
     ci: azure
     template: docker-tests/azure.linux.yml