It all depends on the query itself. If you're doing an indexed lookup (Where indexedColumn = somevalue)or a sort on an indexed column (with no Where clause), then limit 1 will really speed it up. If you have joins or multiple where/order clauses, then things get really complicated really quickly. But the major thing to take away, using "LIMIT 1" will NEVER slow down a query. It will sometimes speed it up, but it will never slow it down.
Now, there is another issue when dealing with PHP. By default, PHP will buffer the entire result set before returning from the query (mysql_query or mysqli->query will only return after all the records are downloaded). So while the query time may be altered little by the limit 1, the time and memory that PHP uses to buffer the result are significant. Imagine each row has 200 bytes of data. Now, your query returns 10,000 rows. That means PHP has to allocate an additional 2mb of memory (actually closer to 10mb with the overhead of php's variable structure) that you'll never use. Allocating memory can be very expensive, so the general rule is only ever allocate what you need (or think you will need). Downloading 10,000 rows when you only want 1 is just wasteful.
Combine these two effects, and you can see why if you want only 1 row, you should ALWAYS use "LIMIT 1".