Is the LIKE operator case-sensitive with MSSQL Server?

It is not the operator that is case sensitive, it is the column itself.

When a SQL Server installation is performed a default collation is chosen to the instance. Unless explicitly mentioned otherwise (check the collate clause bellow) when a new database is created it inherits the collation from the instance and when a new column is created it inherits the collation from the database it belongs.

A collation like sql_latin1_general_cp1_ci_as dictates how the content of the column should be treated. CI stands for case insensitive and AS stands for accent sensitive.

A complete list of collations is available at https://msdn.microsoft.com/en-us/library/ms144250(v=sql.105).aspx

(a) To check a instance collation

select serverproperty('collation')

(b) To check a database collation

select databasepropertyex('databasename', 'collation') sqlcollation

(c) To create a database using a different collation

create database exampledatabase
collate sql_latin1_general_cp1_cs_as 

(d) To create a column using a different collation

create table exampletable (
    examplecolumn varchar(10) collate sql_latin1_general_cp1_ci_as null
)

(e) To modify a column collation

alter table exampletable
alter column examplecolumn varchar(10) collate sql_latin1_general_cp1_ci_as null

It is possible to change a instance and database collations but it does not affect previously created objects.

It is also possible to change a column collation on the fly for string comparison, but this is highly unrecommended in a production environment because it is extremely costly.

select
  column1 collate sql_latin1_general_cp1_ci_as as column1
from table1

All this talk about collation seem a bit over-complicated. Why not just use something like:

IF UPPER(@@VERSION) NOT LIKE '%AZURE%'

Then your check is case insensitive whatever the collation

You have an option to define collation order at the time of defining your table. If you define a case-sensitive order, your LIKE operator will behave in a case-sensitive way; if you define a case-insensitive collation order, the LIKE operator will ignore character case as well:

CREATE TABLE Test (
    CI_Str VARCHAR(15) COLLATE Latin1_General_CI_AS -- Case-insensitive
,   CS_Str VARCHAR(15) COLLATE Latin1_General_CS_AS -- Case-sensitive
);

Here is a quick demo on sqlfiddle showing the results of collation order on searches with LIKE.