Index will be removed
Creating a view with WITH SCHEMABINDING locks the underlying tables and prevents any changes that may change the table schema.
To change the base table you need to remove the SCHEMABINDING option from the objects or drop and recreate these objects once the table has been altered.
To find out if the view is schema bound use below tsql:
select objectproperty(object_id('view_name'), 'IsSchemaBound') as SchemaBound

To remove the with schemabinding option, just alter the view defination :
alter view schemaname.tablename
-- with schemabinding (remove it )
as
select some_column_name
from schema_name.table_name

Then make changes to the table and then alter the view again now with schemabinding
alter view schemaname.tablename
with schemabinding --(adding back )
as
select some_column_name
from schema_name.table_name

from BOL :

SCHEMABINDING :
Binds the view to the schema of the underlying table or tables. When SCHEMABINDING is specified, the base table or tables cannot be modified in a way that would affect the view definition. The view definition itself must first be modified or dropped to remove dependencies on the table that is to be modified. When you use SCHEMABINDING, the select_statement must include the two-part names (schema.object) of tables, views, or user-defined functions that are referenced. All referenced objects must be in the same database.
Views or tables that participate in a view created with the SCHEMABINDING clause cannot be dropped unless that view is dropped or changed so that it no longer has schema binding. Otherwise, the Database Engine raises an error. Also, executing ALTER TABLE statements on tables that participate in views that have schema binding fail when these statements affect the view definition.