To display only the birthdays that occur in the upcoming months, you need to create a custom view. However, if you try to do it by the start date, you won't get the results you expect. Instead, you need to create a filtered table view that only shows items in the Birthday category and with a recurrence containing the names of the month. Separate each month with a comma, so the filter knows you want to OR the words.

Step 1
Outlook 2010, Outlook 2013, Outlook 2016: Switch to the View ribbon, expand Change View button and select Manage views to open the Manage Views dialog.

Outlook 2007 and earlier: Open the Custom Views Organizer dialog, using either the Views menu on the Advanced toolbar or from the View menu.

To use the Views menu toolbar button in any version, show the Advanced toolbar then scroll to the end of the Views list and select Define views.

In Outlook 2003, go to View, Arrange By, Current View, Define view, as shown in the first screenshot.

Outlook 2002 and earlier doesn't have the Arrange By menu, so you'll just go to View, Current View, Define view.

Step 2
Select New to create a completely new view or select a view then Copy to make a copy of a view or choose Modify to modify the selected view. I recommend using Copy or making a new view.

To use automatic formatting, you'll need to use the Table format for the view.

If you create a new view, the Customize view dialog opens, otherwise, select the view you want to customize and click Modify to open the Customize view dialog.

This is the same dialog you'll see if you choose Custom or Customize current view from the context menu when you click on the row of field names at the top of a message list or by selecting from the view menu. Only use these methods to open it if you want to apply the changes to the currently selected folder. If you want the view to apply to all folders using that view, use the Define views menu.

Create a filter to show only birthdays

First you need to create a filter for the months you want to display, unless you want to include all 12 months. (I prefer showing 3 -4 months at a time.) Skip this step if you want to see all birthdays listed.

1. Click the Filter button.
2. On the Advanced tab, select Recurrence Pattern from the Filter dropdown list. (It's under All Calendar items) or type Recurrence Pattern into the field. Select "contains" if it's not already shown in the Condition field.
   
3. Type the name of the months you want to include in the filter in the Value field, separating the words with commas.

Add a second rule for Categories is (exactly) Birthday, otherwise any recurring event will display. If you wanted to include anniversaries and recurring holidays, the 'recurrence field contains yearly' condition would also work.

Update: The Birthday category is no longer automatically added to Birthdays created by Outlook.

Ok your way back to the Customize views dialog. The view is usable as is, but if you want to highlight the birthdays by month, you'll need to add an automatic formatting rule.

Create an Automatic Formatting rule for the view

Click on the Automatic Formatting button. (This was renamed Conditional formatting in Outlook 2010 and newer.)

Click the Add button to create a new rule. Enter a name for the rule and set the font formatting using the Font button.

Click to Customize button to open the Filter dialog.

As with the filter you created earlier that applies to the folder, create a rule for recurrence pattern contains but only enter the name of one month.

Repeat for each month in your view.

Now when you apply the view to your folder, you can see the upcoming birthdays and they are highlighted in different colors by month.

Because Outlook's recurrence pattern filter doesn't accept natural language words, like "next month", you can't make one view that continually updates. Instead, you'll need to make one view for each month or to reduce the number of views needed, use autoformatting to color the birthdays in each month so it's easier to see which birthdays are in each month.