Pi Day, Pi Minute, or Pi Second?
Javascript (ES6), 114 112 - 15 = 97 bytes

x=&gt;['Pi Day','Pi Minute','Pi Second'].find((x,i)=&gt;[/ar 14/,/(03|15):14:/,/03:14/][i].test(Date()))||'No Pi Time'


Ungolfed:

x=&gt;
['Pi Day', 'Pi Minute', 'Pi Second']  // array of outputs
.find(                                // find first element in the array
    (x, i)=&gt;                          // which returns truthy for this function
    [/ar 14/, /(03|15):14:/, /03:14/] // array of regex patterns
    [i]                               // get corresponding regex based on index
    .test(Date())                     // test it against current date, date is automatically cast to string
) || 'No Pi Time'                     // if no result, then return "No Pi Time"


Thanks for -2 bytes @edc65