There is no online documentation provided by Esri indicating that ArcGIS Online Map Viewer does not support the option for label placement for line features.
This issue is not reproducible in ArcGIS Online Map Viewer. Currently, labeling line features in ArcGIS Online Map Viewer provides three options for placement: Above along, Center along, and Below along, all of which function properly with line features. If the issue occurs, please contact Esri Support Services as additional information is required for further investigation.