First, please check the list you are selecting: Pinned lists such as "recently viewed contacts" and "all contacts" will not have a button. 

With some installations, the Begin PhoneBurner Dial Session button may not initially appear from the Contacts or Leads views in your Salesforce account. Follow these steps to manually add this button to your pages in Salesforce:

Adding the button to the Contacts tab in Salesforce (Classic):

1. Click the Setup link at the top right
2. Expand the 'Customize' option on the left, and go down to 'Contacts'
3. From under 'Contacts', go to 'Search Layouts'
4. Click the "Edit" link next to 'search results' and also 'Contacts List View'
5. Move the Begin PhoneBurner Session button from the Available Buttons field on the left to the field on the right, then click save. 

Adding the button to the Leads tab in Salesforce (Classic):

1. Click the Setup link at the top right
2. Expand the Customize option on the left, and go down to Leads
3. From under Leads, go to Search Layouts
4. Click the Edit link next to 'search results' and also 'Leads List View'
5. Move the Begin PhoneBurner Session button from the Available Buttons field on the left to the field on the right, then click save.

Adding the button to the Leads and Contacts tab in Salesforce (Lightning):

1. Click Setup link at the top right
2. Click Object Manager towards the top middle
3. Click Contact or Leads (in alphabetical order) Do this for both.
4. Click Search Layouts on the lefthand side
5. Click Edit for Default Layout  
6. Move the button Begin PhoneBurner Session to the Selected Buttons field and click save.  

7.  Click "search layouts for classic view"  and edit on the list view and repeat the step above.  NOTE: Even if you are only using lightning, you must add it here as well! 

In Some versions it may look like this. 

For information on how to add the Begin Contact Dial session button in your opportunities object, click here. 

For information on how to add it to the Task object, See: