When are computed columns computed?
This is very easy to prove on your own. We can create a table with a computed column that uses a scalar user-defined function, and then check plans and function stats before and after both an update and select, and see when an execution gets recorded.
Let's say we have this function:
CREATE FUNCTION dbo.mask(@x varchar(32))
RETURNS varchar(32) WITH SCHEMABINDING
AS
BEGIN
  RETURN (SELECT 'XX' + SUBSTRING(@x, 3, LEN(@x)-4) + 'XXXX');
END
GO

And this table:
CREATE TABLE dbo.Floobs
(
  FloobID int IDENTITY(1,1),
  Name varchar(32),
  MaskedName AS CONVERT(varchar(32), dbo.mask(Name)),
  CONSTRAINT pk_Floobs PRIMARY KEY(FloobID),
  CONSTRAINT ck_Name CHECK (LEN(Name)&gt;=8)
);
GO

Let's check sys.dm_exec_function_stats (new in SQL Server 2016 and Azure SQL Database) before and after an insert, and then after a select:
SELECT o.name, s.execution_count
FROM sys.dm_exec_function_stats AS s
INNER JOIN sys.objects AS o
ON o.[object_id] = s.[object_id]
WHERE s.database_id = DB_ID();

INSERT dbo.Floobs(Name) VALUES('FrankieC');

SELECT o.name, s.execution_count
FROM sys.dm_exec_function_stats AS s
INNER JOIN sys.objects AS o
ON o.[object_id] = s.[object_id]
WHERE s.database_id = DB_ID();

SELECT * FROM dbo.Floobs;

SELECT o.name, s.execution_count
FROM sys.dm_exec_function_stats AS s
INNER JOIN sys.objects AS o
ON o.[object_id] = s.[object_id]
WHERE s.database_id = DB_ID();

I see no function call on the insert, only on the select.
Now, drop the tables and do it again, this time changing the column to PERSISTED:
DROP TABLE dbo.Floobs;
GO
DROP FUNCTION dbo.mask;
GO

...
  MaskedName AS CONVERT(varchar(32), dbo.mask(Name)) PERSISTED,
...

And I see the opposite happening: I get an execution logged on the insert, but not on the select.
Don't have a modern enough version of SQL Server to use sys.dm_exec_function_stats? No worries, this is captured in the execution plans, too.
For the non-persisted version, we can see the function referenced only in the select:




While the persisted version only shows the computation happening on insert:




Now, Martin brings up a great point in a comment: this isn't always going to be true. Let's create an index that doesn't cover the persisted computed column, and run a query that uses that index, and see if the lookup gets the data from the existing persisted data, or computes the data at runtime (drop and re-create function and table here):
CREATE INDEX x ON dbo.Floobs(Name);
GO

INSERT dbo.Floobs(name) 
  SELECT LEFT(name, 32) 
  FROM sys.all_columns 
  WHERE LEN(name) &gt;= 8;

Now, we'll run a query that uses the index (actually it uses the index by default in this specific case anyway, even without a where clause):
SELECT * FROM dbo.Floobs WITH (INDEX(x))
  WHERE Name LIKE 'S%';

I see additional executions in function stats, and the plan doesn't lie:



So, the answer is IT DEPENDS. In this case, SQL Server thought it would be cheaper to re-compute the values than to perform lookups. This could change due to a variety of factors, so don't rely on it. And this can happen in either direction whether or not a user-defined function is used; I only used it here because it made it that much easier to illustrate.