Why is Romeo so sad?
Romeo's emotions are a rollercoaster through the play: when we first hear of him in Act 1 scene 1 he is love-sick over Rosaline and "Many a morning hath he there been seen / With tears augmenting the fresh morning's dew" and is sick because he is "Out of her favour where I am in love". He speaks almost entirely in this scene in rhyming couplets which may be being used by Shakespeare to either ennoble his feelings or, in my opinion more likely, to alert the reader to their artificiality.
In contrast, when he first sees Juliet in Act 1 scene 5, he and Juliet share a sonnet which Shakespeare most certainly uses to indicate a genuine and true connection.
Thereafter, Romeo is emboldened by his love for Juliet, allowing him to enter the Capulet garden, to belittle the dangers there which Juliet is painfully aware of.
After Act 3 scene 1 where Romeo kills Tybalt, true all-consuming grief encompasses Romeo as it does with Juliet. Romeo now banished on pain of death, their marriage seems destined never to be consummated and they both contemplate suicide. I think Shakespeare intends us to assume that they are genuinely suicidal, as opposed to traditional courtly protestations.
Finally, after a litany of mistakes, Romeo is convinced that Juliet is dead and is heartbroken, taking poison so as to remain with her in death as he could not in life.
Romeo is heartsick because while he can love Juliet, it's not in the way that he wants it to be. He wants to be able to shout his love for her from the rooftops instead of having to keep it secret. Romeo wants the blessings from their families for a long and happy life together, not disapproval and disdain of their love because of past conflict involving their families that he and Juliet weren't even a part of.
In the end, he is devastated because he thinks Juliet is dead. His heart feels broken and any possibility for a future with her he believes is gone.