From the precise moment we meet Sylphiel we know she feels something for Gourry. While her feelings are strong, she is way too shy to express them in ways Gourry can comprehend. As I mentioned before, she knew Gourry from a long time ago and had developed a hero-worship type crush on him. I use the term "crush" loosely because it may have started as one when she first met him, but it had possibly developed into love over time. Absence does make the heart grow fonder after all.
Sadly, Gourry appears to be completely oblivious to Sylphiel's unspoken affection and in the end he probably doesn't feel the same way, he may just see her as a sister. He could only remember her name in the first place just because she was such a great cook although Sylphiel doesn't seem to mind that fact. While Gourry seems unaware to her love, Lina can see it plain as day. Lina herself cares deeply for Gourry, but she is unsure what her emotions are or what she feels. She does know that Sylphiel and her gestures of love bother her. Lina just never considered the fact that Gourry might leave her for someone like Sylphiel. And on the reverse, Sylphiel is afraid that Lina and Gourry's friendship has taken the place of the friendship she had with Gourry and he no longer needs her.
Gourry is kind to Sylphiel and protects her, he even comforted her when her father died. When he complained about Lina to her in such a carefree manner though, it's then that Sylphiel realized that she couldn't be in his heart the way she wanted because someone had already taken that position. It is true Sylphiel tried to compete with Lina for Gourry's affection by learning the Dragon Slave but because she knew the two loved each other she gave him up at the end, even though it ruined her chances of ever being with him. This does not hurt Sylphiel, nor does she resent Lina for it, instead she goes to the sidelines and lets them be, which is a great act of selflessness and love. That shows that she truly loves Gourry, for she only wants him to be happy and it's with Lina he would be happiest with and not her. So she is content to see them together and she makes sure that nothing will come between them.
There are some really cute and sweet moments between them and you can tell Sylphiel cares deeply for him but it wouldn't work, and not just because of Lina being in the picture, but because Gourry seems to not want the same things as Sylphiel does in a relationship and he doesn't know what Sylphiel needs nor wants. Even though it broke her heart, she gave up pursuing Gourry's affections when she realized that he was in love with someone else. Sylphiel would never try to tell him how she feels about him because she wouldn't want him to feel guilty. Deep down I think that she would just want him to be happy even if it was at the cost of her own happiness.