Happiness-all-the-time is an intricate and quite militant art-form. Something that requires a high degree of 'self-policing'. If you manage to do it - to actually persist in the good feeling - no matter what - you would have sufficiently developed the skillsets of:

1 - self-awareness and persistently gaining greater awareness over previous blindspots (being mindful of what you really feel, not just glossing over negative feelings),

2 - detachment (not falling into the givenness of immediate thoughts and feelings about something), so learning to step back from the appearance of present conditions,

3 - imagination & revision - the art and muscle of mentally shifting thoughts to their happy version, which requires the mastery of the steps above.

As life unfolds, you will automatically be faced with Limiting Beliefs that will reveal themselves as thoughts that feel uncomfortable, because they are pitted against your desires. You cannot simultaneously hold worrying thoughts and also happy thoughts about a person, place or thing.

So, your limiting beliefs will continue to crop up and challenge you to resolve them, to develop your abilities to persisting to stay happy, and stay internally committed to the version of things as you desire them to be. And as you persistently redirect thoughts about the different things that you desire, you will chip away at your previously limiting beliefs, and develop new belief habits.

Desired Manifestation will then follow suit, most fundamentally in the sense that you WILL ALWAYS GET WHAT YOU WANT IMMEDIATELY (the feeling or sense of your desire fulfilled), and the echoes of it will then emanate out to return to you in physical form. This means that if the universe is a reflection of you, and you persistently hold yourself in the state of happiness, you will ripple out happiness and be on the receiving end of all the things that make you happy, which includes your DESIRES MANIFESTING.

So you see, being persistently HAPPY or what Abraham Hicks would term staying 'in the Vortex', is not that simple. But if you take it on as a very serious task, it will be the battering ram to break down the door to your Internal Mansion, to a life of HEAVEN on EARTH, and to always getting everything that you want.

An underlying premise in your question is that a belief is some "thing"–there is some truth to that. But, if you understand what that "thing" really is, the answer to your question becomes quite simple.

A belief is a habit of thought. A thought you are accustomed to thinking. When you are "happy all the time," you are no longer thinking thoughts representative of those "unhappy" beliefs, thereby giving way to new habits of thought. Those new habits of thought essentially form new beliefs that "overwrite" the old ones. The old situations, memories, feelings, etc. that would trigger the thoughts associated with your old beliefs begin to come up less, or are simply replaced by new better-feeling thoughts.

When you drain limiting beliefs of their mental airtime, your brain stops prioritizing those thoughts and allows those neural pathways to fade. That's what's happening when you are happy all the time. In time, your old limiting thoughts are no longer a habit, and as a corollary, your limiting beliefs have been released, without you having to "do" anything.

I've worked with the "be happy" manifestation idea a few times and despite my own logical resistance to it, something awesome has happened every time I played this game. Some ideas on why this works:

1) You have a deliberate decision behind this process that it will work and so it does.

2) Being happier gets you to stop putting attention and awareness on your limiting beliefs. You stop giving a damn about them, stop using them as an excuse, so they no longer have so much power in your life.

3) I think we often fear our limiting beliefs because of how unhappy they make us so we try hard to fight against them. But what if you just decide to be happy and get completely absorbed in your own fulfillment and bliss (for its own sake)? You'll see that your limiting belief no longer hurt you because you decided to switch your awareness, to change your self-concept. And that knocks the power of limiting beliefs down a peg, you'll begin feeling like it's not as powerful as before.

4) And as you identify more as a happier person it becomes your subconscious self-concept and has more momentum behind it than your previous limiting beliefs. You'll find yourself automatically thinking as if you are happy, fulfilled person and your new thinking will break the mold of the previous limiting beliefs.

5) As the Vortex draws you in, you may be inspired to pursue actions and ideas that help you let go of limiting beliefs.