Are the words in bolds necessary?

[flexnow]

1. We can not afford (to buy) a new car.
2. I'm really looking forward to (starting) my new job.
3. He denies (to) having seen here before.

My guess: 1.No  2.No  3.Yes
Thanks for your help

[tiorthan]

Nope. Number 3 is the only one where I can say with certainty: Burn it with fire! It doesn't belong there.
After "deny" you need an ing-form.
For 1 and 2. You can leave the marked words out, but that changes the meaning.

[Alyssea]

Leaving the words out on 1 and 2 doesn't change the meaning. They're implied if you leave them out. As for 3, yeah, definitely don't put "to" in there, but that sentence isn't very good either way. "He denies having been seen here before" or "He denies having seen this place before" would be better, depending on the meaning you're going for. 

[tiorthan]

They're implied if you leave them out.

But that is not the same as having the same meaning. The difference between implication and information is a major source of misunderstandings between humans, and is also the reason why we have lawyers.

As for 3, yeah, definitely don't put "to" in there, but that sentence isn't very good either way. "He denies having been seen here before" or "He denies having seen this place before" would be better, depending on the meaning you're going for. 

I think that was supposed to read "her" and not "here".

[Alyssea]

Ah, yeah, "her" also makes more sense. 
And, in this case, they're implied to the point of being obvious. Similar to a sentence like "I know that it will be fine" vs. "I know it will be fine." Meaning doesn't change at all. With "we cannot afford a new car," there's literally no other way to interpret it besides buying. Adding "to buy" is extraneous. Some implied things do have other potential meanings, but neither of the examples do. 

[tiorthan]

With "we cannot afford a new car," there's literally no other way to interpret it besides buying.

That's where I disagree.

The ambiguity begins at "afford". That word does not necessarily refer to financial affordability. We conclude that based on our interpretation of the thing that cannot be afforded which is "a new car" and our experience tells us, that people mean buying when they talk about not being able to afford a new car.

But in a different context, that sentence can have a different meaning. There are communities that are opposed to certain aspects of modern capitalism. A person in such a community, even if they were financially able to buy a new car (reas new as in modern) or if they would receive a new car as a gift could not afford to own a new car without being shunned by said community.

Of course, you are right when you say that both sentences have the same meaning. If no context is given, we have to assume the most likely context given the cultural background of the average speaker of a language. That is a skill you do not have to teach to anyone, because humans do it automatically.

Similar to a sentence like "I know that it will be fine" vs. "I know it will be fine." Meaning doesn't change at all.

That case is a bit different though. Here different grammatical forms are used but the content stays the same. From the point of view of a linguist, you're not even leaving a word out here you just replace "that" by a zero-relative-pronoun (linguists are weird people).

I do understand, that the brief note on changed meaning I gave in my first post could be misinterpreted (as pointing towards a radical change in meaning). So I guess I have to be more precise in the future.

[flexnow]

 Thank you guys for your sophisticated answers 
Yes, I did indeed mean 'her' not 'here'. Sorry for that 

[Alyssea]

With "we cannot afford a new car," there's literally no other way to interpret it besides buying.

That's where I disagree.

The ambiguity begins at "afford". That word does not necessarily refer to financial affordability. We conclude that based on our interpretation of the thing that cannot be afforded which is "a new car" and our experience tells us, that people mean buying when they talk about not being able to afford a new car.

But in a different context, that sentence can have a different meaning. There are communities that are opposed to certain aspects of modern capitalism. A person in such a community, even if they were financially able to buy a new car (reas new as in modern) or if they would receive a new car as a gift could not afford to own a new car without being shunned by said community.

I hadn't thought of it that way; that is a reasonable alternate meaning. 

     
Of course, you are right when you say that both sentences have the same meaning. If no context is given, we have to assume the most likely context given the cultural background of the average speaker of a language. That is a skill you do not have to teach to anyone, because humans do it automatically. 

I do understand, that the brief note on changed meaning I gave in my first post could be misinterpreted (as pointing towards a radical change in meaning). So I guess I have to be more precise in the future.

I'd just say that adding "to buy" and "starting" clarifies the sentences, but doesn't actually change their meaning, since the default assumed meaning would be the same meaning that the sentences have when those words are added. If someone were wanting to give the meaning you described above, then adding words to clarify it in that case would change the meaning of the sentence.