I was studying some particles and I already understood the difference between of が and は. Also the function of the particle を and its usage. I wonder if it is possible to NOT use it in a phrase? Take a look at the example below.
ねこがこいぬをいじめていた。- A cat was bullying a puppy
Do we really need that mora を after the puppy object there?
Ok let me give a proper answer. First of all, I think you have a typo in your example since いじのてた does not mean anything and I believe you probably mean いじめていた from the verb 苛｛いじ｝める that means "to tease, to torment, to persecute" etc.
EDIT 2: After all the comments, I will re-organize my answer trying to address all the points in hope everyone will be happy with it. In particular there was a stupid mistake I will remove.
I will cut it short and say that the answer to your question is, in my opinion: it depends (hence no, you don't always need を after the object).
Take this simple example: I can speak Japanese, that is:
（私は）日本語を話せます。
In this case you need を.
However, if you are casually talking with someone you would say:
（私は）日本語話せるよ。
This second one, in the right context, is also acceptable. If you were talking in a casual register with your friends but stuffed all the particles at the right place it would probably sound a bit weird. However, do not confuse grammar with formal/informal speech. Grammar would want you to put that particle there. In general in written text you always must put the particles where needed. However, I think that since we live in an era where instant messaging led to an extreme increase of "informal" text, it's also important to mention that even in "written text" depending on the context it is acceptable to omit を (as well as all other particles). Again, bear in mind this is not grammatical.
Now the trick. If you simply ask about placing を after the object, then certainly you do not need always を. Consider the following:
私は日本語が話せます。
私は日本語も話せます。
These also make perfect sense! For example, if you want to express that Japanese is the language you can speak, among others that you can't you would use が. If you are saying that you speak Japanese as well as other languages, you would use も. These are two examples of sentences where the object is not followed by を.
There is another important example that came to mind thinking of this thoroughly (and thanks to all the comments that made me do so). That is:
You can omit を when you change the structure of the sentence making what the particle を is indicating the "topic" of the sentence. What does this mean?
I could use the same example from above but let's make a different one:
（私は）コンビニでこの弁当{べんとう}を買{か}いました.
I (subject) bought this bento (object) at the convenience store.
Now make the bento the topic of the sentence, that is:
この弁当はコンビニで買いました.
Notice that the bento is still the object here (although omitted you, who are buying, are the subject of the sentence).
This is another example where を after an object is omitted, or better replaced by は. Maybe it just follows in the category above where simply another particle is used but I think could be interesting to point out as well.
Bottom line: not necessarily the object of a sentence is always followed by を. You can indeed use other particles depending on the context and what you actually want to express, or omit it completely if you are having a casual conversation (whether oral or written).
