How do I drop a foreign key constraint only if it exists in sql server?

This is a lot simpler than the current proposed solution:

IF (OBJECT_ID('dbo.FK_ConstraintName', 'F') IS NOT NULL)
BEGIN
    ALTER TABLE dbo.TableName DROP CONSTRAINT FK_ConstraintName
END

If you need to drop another type of constraint, these are the applicable codes to pass into the OBJECT_ID() function in the second parameter position:

C = CHECK constraint
D = DEFAULT (constraint or stand-alone)
F = FOREIGN KEY constraint
PK = PRIMARY KEY constraint
UQ = UNIQUE constraint

You can also use OBJECT_ID without the second parameter.

Full List of types here:

Object type:

AF = Aggregate function (CLR)
C = CHECK constraint
D = DEFAULT (constraint or stand-alone)
F = FOREIGN KEY constraint
FN = SQL scalar function
FS = Assembly (CLR) scalar-function
FT = Assembly (CLR) table-valued function
IF = SQL inline table-valued function
IT = Internal table
P = SQL Stored Procedure
PC = Assembly (CLR) stored-procedure
PG = Plan guide
PK = PRIMARY KEY constraint
R = Rule (old-style, stand-alone)
RF = Replication-filter-procedure
S = System base table
SN = Synonym
SO = Sequence object

Applies to: SQL Server 2012 through SQL Server 2014.

SQ = Service queue
TA = Assembly (CLR) DML trigger
TF = SQL table-valued-function
TR = SQL DML trigger
TT = Table type
U = Table (user-defined)
UQ = UNIQUE constraint
V = View
X = Extended stored procedure